Algorithms for Representation Theory of Real Reductive Groups

Jeffrey Adams and Fokko du Cloux

July 18, 2008

Introduction

The irreducible admissible representations of a real reductive group such as GL(n, ℝ) have been classified by work of Langlands, Knapp, Zuckerman and Vogan. This classification is somewhat involved and requires a substantial number of prerequisites. See [13] for a reasonably accessible treatment. It is fair to say that it is difficult for a non-expert to understand any non-trivial case, not to mention a group such as E_{8}.

The purpose of these notes is to describe an algorithm to compute the irreducible admissible representations of a real reductive group. The algorithm has been implemented on a computer by the second author. This work is part of the Atlas of Lie Groups and Representations project. An early version of the software (Version 0.3 as of July 2008), and other documentation and information, may be found on the web page of the Atlas

project, www.liegroups.org.

Here is some more detail on what the algorithm and the software do:

(1) Allow the user to define

(a) A complex reductive group G,

(b) An inner class of real forms of G,

(c) A particular real form G(ℝ) of G.

2000 Mathematics Subject Classification: Primary 22E50, Secondary 05E99

This work was supported in part National Science Foundation Grants #DMS-0554278 and #DMS-0532393.

(2) Enumerate the Cartan subgroups of G(ℝ), and describe them as real tori,

(3) For any Cartan subgroup H(ℝ) compute W (G(ℝ), H(ℝ)) (the “real” Weyl group),

(4) Compute a set X parametrizing the set K\G/B where B is a Borel subgroup and K is the complexification of a maximal compact subgroup of G(ℝ),

(5) Compute a set Z parametrizing the irreducible representations of G(ℝ) with regular integral infinitesimal character,

(6) Compute the cross action and Cayley transforms on X and Z,

(7) Compute Kazhdan-Lusztig polynomials.

In fact the proper setting for all of the preceding computations is not a single real group G(ℝ), but an entire inner class of real forms, as described in Sections 2–4.

The approach used in these notes most closely follows [1]. This reference has the advantage over [3], which later supplanted it, in that it focuses on the case of regular integral infinitesimal character, and avoids some complications arising from the general case. There are a few changes in terminology from these references which are discussed in the remarks. We also make extensive use of [4]. This work has some overlap with, and was influenced by, that of Richardson and Springer [21].

A number or examples illustrating the algorithm and the software may be found in Guide to the Atlas Software: Computational Representation Theory of Real Reductive Groups [2]. The reader is encouraged to use the software, and consult [2], while reading this paper. An earlier version of this paper by the second author is available on the Atlas web site [9].

The authors thank David Vogan for his numerous contributions to this project, and Marc van Leeuwen, Patrick Polo and the referee for very helpful suggestions.

Fokko du Cloux died of ALS in November 2007. Fokko played a major role in development of the algorithm described in this paper, in addition to writing the atlas software. This work began in 2004 and was mostly complete by late 2006. Fokko was an active participant in the writing of this paper, which was originally submitted shortly after his death.

Index of Notation

Aut(G), Int(G), Out(G): the (holomorphic) automorphisms of G, the inner automorphisms, and the outer automorphisms Aut(G)/Int(G), respectively (Section 2.1).

B: set of Borel subgroups of G (Section 8), in bijection with G/B.

(B, H, {X_{α}}): splitting or pinning data for G (Section 2.1). Defines a splitting Out(G) → Aut(G) (cf. (2.8)).

c^{α}, c_{α}: noncompact imaginary and real Cayley transforms, respectively (Definitions 14.1 and 14.8).

D = (X, Δ, X^{∨}, Δ^{∨}): root datum defining a complex connected reductive group (2.1).

D(G, H): root datum associated to a complex connected reductive group G and a Cartan subgroup H (2.5).

D_{b} = (X, Π, X^{∨}, Π^{∨}), D_{b}(G, B, H): abstract based root datum, and the based root datum associated to G, H and a Borel subgroup B (2.6).

D^{∨}, D_{b}^{∨}, G^{∨}: dual (based) root datum and dual group associated to D, D_{b} and G respectively (end of Section 2).

(G, γ): basic data consisting of a complex connected reductive group G and an involution γ ∈ Out(G) (Section 4).

G^{Γ}: given basic data (G, γ), the extended group containing G; G^{Γ} = G ⋊ Γ where Γ = {1, σ} and σ acts by a distinguished involution mapping to γ (Definition 5.1).

(G^{∨}, γ^{∨}): basic data dual to (G, γ) (Section 4).

H′_{−τ}, H_{−τ}, A_{τ}, T_{τ}: various subgroups of the Cartan subgroup H, depending on a twisted involution τ ((11.1) and Remark 11.3). In particular A_{τ} ⊂ H_{τ} ⊂ H′_{τ}, A_{τ} and T_{τ} are connected complex tori, and H = T_{τ}A_{τ}.

I(G, γ): set of strong involutions in G^{Γ} (Definition 5.5).

I: {ξ_{i} | i ∈ I} is a set of representatives of the set of strong real forms, i.e. I(G, γ)/G (5.15)(a).

I_{i}: for i ∈ I, strong involutions conjugate to ξ_{i} (5.15).

I_{W}: twisted involutions in W = {τ ∈ W^{Γ}\W | τ^{2} = 1} (9.11)(h) and (9.14).

L, L_{c}: equivalance classes of (complete) two-sided L-data for (G, γ) (7.14).

N, N^{Γ} : normalizer of H in G and G^{Γ}, respectively ((9.11)(a)).

p̃, p: maps from X̃ and X to I_{W}, respectively ((9.11)(i)).

P, P^{∨}, P_{reg}, P_{reg}^{∨}, P (G, H), P^{∨}(G, H): weight and coweight lattices and their regular elements (Section 4).

P, P_{c}: equivalence classes of (complete) one-sided L-data for (G, γ)) (7.9); P = (I × B)/G ((8.1)(a)). Also P^{∨}, P_{c}^{∨} on the dual side.

P[ξ]: set of equivalence classes of one-sided L-data (ξ′, B) such that ξ is conjugate to ξ ((8.1)(b)). In bijection with K_{ξ}\G/B ((8.1)(e)).

S = (ξ, B): one-sided L-datum for (G, γ), consisting of of a strong involution ξ and a Borel subgroup B of G (Definition 7.6). Also S^{∨} = (η, B_{1}^{∨}) is a one-sided L-datum for (G^{∨}, γ^{∨}).

S_{c}^{∨} = (η, B_{1}^{∨}, λ): complete one-sided L-datum for (G^{∨}, γ^{∨}), consisting of an L-datum (η, B_{1}^{∨}) and λ ∈ 𝔥^{∨} satisfying exp(2πiλ) = η^{2} (Definition 7.1).

S = (S, S^{∨}): two-sided L-datum, where S, S^{∨} are L-data for (G, γ) and (G^{∨}, γ^{∨}) respectively, satisfying an additional compatibility condition (Definition 7.10).

S_{c} = (S, S_{c}^{∨}): complete two-sided L-datum, where S is an L-datum for (G, γ), S_{c}^{∨} is a complete L-datum for (G^{∨}, γ^{∨}), satisfying an additional compatibility condition (Definition 7.10).

W, W^{Γ}: W = N/H is the Weyl group, and W = N^{Γ}/H is the extended Weyl group (9.11)(b).

W̃: the Tits group; a subgroup of G mapping onto W (Definition 15.1).

X = X (G, γ): the one-sided parameter space = X̃/H = {ξ ∈ Norm_{G^{Γ}\G}(H) | ξ^{2} ∈ Z(G)}/H (Definition 9.2).

X^{r} = X^{r}(G, γ): the reduced parameter space; a finite subset of X (Definition 13.4).

X̃: strong involutions normalizing H = {ξ ∈ Norm_{G^{Γ}\G}(H) | ξ^{2} ∈ Z(G)} (9.11)(d).

X_{τ}, X̃_{τ}: for τ a twisted involution, the fiber of the map p : X → I_{W} (resp. p̃: X̃ → I_{W}) over τ (beginning of Section 11). X [x]: for x ∈ X, subset of X consisting of elements conjugate to x (9.7). X (z): for z ∈ Z(G) the elements of X satisfying x^{2} = z ((9.13)(c)). X_{τ}(z): for z ∈ Z(G), τ ∈ I_{W}, X_{τ}(z) = X_{τ} ∩ X (z); the elements x ∈ X satisfying p(x) = τ and x^{2} = z (beginning of Section 11). Z = Z(G, γ): the two-sided parameter space, contained in X (G, γ)×X (G^{∨}, γ^{∨}) (Definition 10.1). δ: distinguished element of X or I_{W} (Definition 5.1 and before Lemma 9.12). Δ_{i}, Δ_{r}, Δ_{cx}, W_{i}, W_{i,τ}, W_{r}, W_{r,τ}, etc.: various root systems and Weyl groups (beginning of Section 12). γ^{∨}: element of Out(G^{∨}) corresponding to γ ∈ Out(G) (Defintion 2.11). θ_{ξ}, K_{ξ}: for ξ ∈ X̃, θ_{ξ} = int(ξ), an involution of G, and K_{ξ} = G^{θ_{ξ}} (Definition 5.5). Thus θ_{ξ} is the Cartan involution of a real form of G, and K_{ξ} is the corresponding complexified maximal compact subgroup (Section 3). θ_{i}, K_{i}: For i ∈ I, Cartan involution and complexified maximal compact subgroup associated to ξ_{i} (5.15). ξ : strong involution (with respect to (G, γ)); ξ ∈ G^{Γ}\G satisfies ξ^{2} ∈ Z(G) (Definition 5.5). Then θ_{ξ} = int(ξ) is an involution of G. (ξ, π): Harish-Chandra module for a strong involution; a strong involution ξ and a (𝔤, K_{ξ})-module π. A Harish-Chandra module for a strong real form is a G-orbit of such pairs (Section 6). Π(G(ℝ), λ): set of equivalence classes of irreducible admissible representations of G(ℝ) with infinitesimal character λ (Definition 1.1). Π(G(ℝ), Λ): for Λ ⊂ P_{reg}, ∐_{λ∈Λ} Π(G(ℝ), λ) (before Theorem 7.17). Π(G, ξ): equivalence classes of (𝔤, K_{ξ})-modules with regular integral infinitesimal character (Definition 6.3). Π(G, γ): representations of strong real forms of G with regular integral infinitesimal character; = {(ξ, π) | ξ ∈ I, π ∈ Π(G, ξ)}/G (Definition 6.3). Π(G, γ, Λ): for Λ ⊂ P_{reg}, the subset of Π(G, γ) consisting of representations with infinitesimal character contained in Λ.

Π_{ϕ}(G, ξ): L-packet of representations of strong real form ξ associated to the L-homomorphism ϕ (after 6.5).

Π_{ϕ}(G, γ): large L-packet; union of L-packets for all strong real forms (6.6).

Π_{S_{c}^{∨}}(G, γ): large L-packet = Π_{ϕ}(G, γ) where ϕ is defined by (7.3) (cf. (7.5)).

1 Overview

The primary aim of this paper is to distill a well-known but difficult theory into an algorithm which can be implemented on a computer. While the resulting algorithm is self-contained and comparatively elementary, even understanding the algorithm itself requires a fairly deep knowledge of the mathematics. In this section we give a high level overview of the algorithm, before going into more detail in the remainder of the paper. An outline of the contents of the paper appears at the end of this Overview (Section 1.7).

We assume the reader is familiar with the theory of admissible representations of real reductive groups. A good introduction is Knapp’s book [13].

In this section we write G(ℂ) for a complex reductive group, with real points G(ℝ). We will make various simplifying assumptions in the course of this overview; this is for ease of exposition, and the general statements will be found in the body of the paper. See the end of this section for a discussion of the issues involved.

Let 𝔤 be the Lie algebra of G(ℂ). Fix a Cartan involution θ of G(ℂ) corresponding to G(ℝ). Thus K = G(ℝ)^{θ} is a maximal compact subgroup of G(ℝ). The basic goal is to parametrize the irreducible admissible representations of G(ℝ), or equivalently the irreducible admissible (𝔤, K)-modules. This is an infinite (typically uncountable) set.

Suppose H(ℂ) is a Cartan subgroup of G(ℂ), and let 𝔥 = Lie(H(ℂ)). By the Harish-Chandra homomorphism, associated to λ ∈ 𝔥^{∗} is an infinitesimal character which we also denote λ. We say λ is regular (resp. integral) if ⟨λ, α^{∨}⟩ ≠ 0 (resp. ∈ ℤ) for all roots α of 𝔥 in 𝔤.

Since we will be working with different Cartan subgroups simultaneously, it is convenient to fix one, denoted H_{a}(ℂ), the abstract Cartan subgroup. Then we fix once and for all λ ∈ 𝔥_{a}^{∗}. The first basic reduction is to consider representations with infinitesimal character λ.

Definition 1.1 Fix λ ∈ 𝔥_{a}^{∗}, and let Π(G(ℝ), λ) be the set of irreducible admissible representations of G(ℝ) with infinitesimal character λ.

By a result of Harish-Chandra this is a finite set. The main result of this paper is an algorithm to compute Π(G(ℝ), λ) when λ is regular and integral.

There are a number of approaches to classifying Π(G(ℝ), λ). We focus on three of them, which are intertwined with each other, and each of which plays a role in the algorithm.

1.1 The Langlands classification

Many representations of reductive groups can be constructed using characters of Cartan subgroups. For example if G is a reductive group over a finite field these are the representations R_{T} (θ) of Deligne and Lusztig [8]. For generic θ R_{T} (θ) is irreducible.

For another example, suppose 𝔽 is a finite or local field, G = G(𝔽) is split, and B = HN is a Borel subgroup of G. If χ is a character of H then Ind_{B}^{G}(χ⊗1) is a minimal principal series representation of G; again for generic χ this is irreducible.

For a final example assume G(ℂ) is semisimple and simply connected, and G = G(ℝ) is a real form of G(ℂ), containing a compact Cartan subgroup T. Suppose χ is a character of T such that ⟨dχ, α^{∨}⟩ ≠ 0 for all roots α. Associated to χ is the discrete series representation of G with Harish-Chandra parameter dχ, and every discrete series representation of G is of this form.

The Langlands classification for G(ℝ) is built out of the second two cases. For now we assume that G(ℂ) is acceptable, i.e. ρ (one half the sum of a set of positive roots) exponentiates to H_{a}(ℂ). This holds for example if G(ℂ) is semisimple and simply connected.

Consider the set of pairs (H(ℝ), χ) where H(ℝ) is a Cartan subgroup of G(ℝ), χ is a character of H(ℝ), and dχ is G(ℂ)-conjugate to λ. The group G(ℝ) acts on these pairs by conjugation. We define character data as follows:

(1.2)  C(G(ℝ), λ) = {(H(ℝ), χ) | dχ is G(ℂ)-conjugate to λ}/G(ℝ).

Proposition 1.3 (Langlands Classification) Assume G(ℂ) is acceptable and λ is regular. There is a natural bijection

(1.4)  Π(G(ℝ), λ) ⟷^{1−1} C(G(ℝ), λ).

There are many versions of the Langlands classification, for example see [13] or [26]. This restatement of (a special case of) the classification is taken from [4, Theorem 8.29]. Since G(ℂ) is acceptable we do not need the ρ-cover of H, and since λ is regular we can take Ψ to be the set of positive, real integral roots defined by λ (notation as in [4]).

Thus one version of our algorithm would be to replace the right hand side of (1.4) by a computable combinatorial object. Here is an idea of what this involves.

First we fix a set H_{1}(ℝ), …, H_{n}(ℝ) of representatives of the conjugacy classes of Cartan subgroups. An algorithm for computing this set is given by Kostant[14], [15], and independently by Borel (unpublished) and Sugiura [24].

Now fix i and let H(ℝ) = H_{i}(ℝ). Let W (G(ℝ), H(ℝ)) be the “real” Weyl group Norm_{G(ℝ)}(H(ℝ))/H(ℝ). This is a subgroup of the Weyl group W (G(ℂ), H(ℂ)) = W (𝔤, 𝔥). Unlike W (𝔤, 𝔥), W (G(ℝ), H(ℝ)) depends on i, is not necessarily the Weyl group of a root system, and can be somewhat difficult to compute. An algorithm is given in [12]; also see [27, Proposition 4.16].

Lemma 1.5 ([26], Theorem 2.2.4) In the setting of Proposition 1.3 there is a bijection

(1.6)  Π(G(ℝ), λ) ⟷^{1−1} ∐_{i}^{n}(W/W (G(ℝ), H_{i}(ℝ)) × [H_{i}(ℝ)/H_{i}(ℝ)^{0}] .

In particular

(1.7)  |Π(G(ℝ), λ)| = ∑_{i=1}^{n} |W/W (G(ℝ), H_{i}(ℝ))||H_{i}(ℝ)/H_{i}(ℝ)^{0}|.

See [26, Theorem 2.2.4].

The bijection (1.6) depends on a number of choices. In addition to the choice of the H_{i}(ℝ), for each i we have to choose a Borel subgroup containing H_{i}(ℂ), and an embedding of H_{i}(ℝ)/H_{i}(ℝ)^{0} in H_{i}(ℝ). Nevertheless this result gives a good idea of what we need to compute:

(1) the Cartan subgroups H_{1}(ℝ), …, H_{n}(ℝ); and for each i, (2) W (G(ℝ), H_{i}(ℝ)),

(3) H_{i}(ℝ)/H_{i}(ℝ)^{0}.

In any event it is clear that any parametrization of Π(G(ℝ), λ) must (if only implicitly) include a description of items (1-3).

1.2 D-modules

We now describe the classification of Π(G(ℝ), λ) in terms of D-modules. The basic reference is [5], or see [18] for a good introduction to the subject.

Recall θ is a Cartan involution of G(ℂ) corresponding to G(ℝ). Let K(ℂ) = G(ℂ)^{θ}. Then K(ℂ) is a reductive group (possibly disconnected), with real points K = G(ℝ)^{θ}. By the relationship between finite dimensional representations of K and K(ℂ), we may view admissible (𝔤, K)-modules as (𝔤, K(ℂ))-modules.

Let B be the variety of Borel subgroups of G(ℂ), so B = G(ℂ)/B(ℂ) where B(ℂ) is a fixed Borel subgroup. Then K(ℂ) acts on B with finitely many orbits. Let D_{λ} be the sheaf of twisted differential operators on B corresponding to λ. We consider D-module data:

(1.8)  D(G(ℝ), λ) = {(O, τ)}/K(ℂ).

where O is a K(ℂ)-orbit on B, and τ is an irreducible K(ℂ)-equivariant D_{λ}-module.

Here is how to make τ more concrete. Fix x ∈ O and let K_{x}(ℂ) = Stab_{K(ℂ)}(x). Let B(ℂ) be the Borel subgroup of G(ℂ) corresponding to x, and let H(ℂ) be a θ-stable Cartan subgroup contained in B(ℂ). Then τ can be viewed as a character of H(ℝ) whose differential dχ is G(ℂ)-conjugate to λ.

Proposition 1.9 ([5]; [18], Theorem 3.9) There is a natural bijection

(1.10)  Π(G(ℝ), λ) ⟷^{1−1} D(G(ℝ), λ).

It is reasonable to look for an algorithm to compute the right hand side.

Fix an orbit O of K(ℂ) on B. By the Proposition, for every D_{λ}-module τ on O we obtain an irreducible representation. Varying τ we obtain a finite set of representations associated to O. For (not very good) reasons which will become clear in the next section (also see [27, Section 8]) we refer to this set as an R-packet, and denote it Π_{R}(G(ℝ), O, λ). Thus Π(G(ℝ), λ) is a disjoint union of R-packets:

(1.11)  Π(G(ℝ), λ) = ∐_{K(ℂ)\B}Π_{R}(G(ℝ), O, λ).

1.3 The Langlands classification using L-groups

We now consider a version of the Langlands classification in terms of Lgroups. Given our group G(ℂ), with real points G(ℝ), let G^{∨Γ} be the Lgroup of G(ℝ) [6]. Thus G^{∨Γ} = G^{∨}(ℂ)⋊Γ, where G^{∨}(ℂ) is the dual group of G(ℂ), Γ = Gal(ℂ/ℝ), and G^{∨Γ} is a certain semidirect product. For example if G(ℝ) is split this is a direct product.

Let W_{ℝ} be the Weil group of ℝ, i.e. W_{ℝ} = ⟨ℂ^{×}, j⟩ where jzj^{−1} = \overline{z} and j^{2} = −1. We consider the space Hom_{adm}(W_{ℝ}, G^{∨Γ}) of admissible homomorphisms of W_{ℝ} into G^{∨Γ}. These are the continuous homomorphisms such that ϕ(ℂ^{×}) consists of semisimple elements of G^{∨}(ℂ), and ϕ(j) ∉ G^{∨}(ℂ).

The version of the Langlands classification in terms of L-groups says that associated to an admissible homomorphism ϕ is a finite set of irreducible admissible representations G(ℝ), called an L-packet. The L-packet of ϕ only depends on the G^{∨}(ℂ)-orbit of ϕ, and we write Π_{L}(G(ℝ), O^{∨}) for the Lpacket associated to such an orbit O^{∨}. An L-packet may be empty (if G(ℝ) is not quasi-split); non-empty L-packets associated to two orbits are equal if only if the orbits are equal. Finally the admissible dual of G(ℝ) is the disjoint union of L-packets.

The representations in Π_{L}(G(ℝ), O^{∨}) all have the same infinitesimal character, which is determined by O^{∨} (see Section 7). Let Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ) be the admissible homomorphisms for which Π_{L}(G(ℝ), ϕ) has infinitesimal character λ. Thus (compare (1.11)) Π(G(ℝ), λ) is a disjoint union of Lpackets:

(1.12)  Π(G(ℝ), λ) = ∐_{O^{∨}} Π_{L}(G(ℝ), O^{∨})

where the union runs over G^{∨}(ℂ) orbits on Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ).

Suppose an L-packet Π_{L}(G(ℝ), O^{∨}) is non-empty. Some additional data is required to specify a particular representation in Π_{L}(G(ℝ), O^{∨}). Choose ϕ ∈ O^{∨}, let S_{ϕ} be the centralizer in G^{∨}(ℂ) of the image of ϕ, and let 𝕊_{ϕ} = S_{ϕ}/S_{ϕ}^{0}. Roughly speaking Π_{L}(G(ℝ), O^{∨}) should be parametrized by characters of 𝕊_{ϕ}. This leads us to define the set of Langlands data

(1.13) L(G^{∨Γ}, λ) = {(ϕ, χ) | ϕ ∈ Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ), χ ∈ 𝕊̂_{ϕ}}/G^{∨}(ℂ).

Suppose G_{1}(ℝ), G_{2}(ℝ) are two real forms of G(ℂ). It may be that the associated L-groups are isomorphic (we say the real forms are inner to each other if this holds). For example this is the case for all real forms of G(ℂ) if G(ℂ) is semisimple and its Dynkin diagram admits no non-trivial automorphisms. Unlike C(G(ℝ), λ) and D(G(ℝ), λ), L(G^{∨Γ}, λ) should be related to representations of all these real forms.

To make this precise it is convenient to assume that G(ℂ) is adjoint. Let G_{1}(ℝ), …, G_{n}(ℝ) be the inequivalent real forms of G(ℂ) with L-group G^{∨Γ}.

Proposition 1.14 ([1], Theorem 3-2) Assume G(ℂ) is adjoint. There is a natural bijection

(1.15)  ∐_{i=1}^{n} Π(G_{i}(ℝ), λ) ⟷^{1−1} L(G^{∨Γ}, λ).

Again it is reasonable to look for an algorithm to compute the right hand side.

1.4 L-packets and R-packets

Fix a real form G(ℝ) of G(ℂ). Consider for the moment the problem of explicitly parametrizing Π(G(ℝ), λ) using L-homomorphisms.

Recall (1.12) Π(G(ℝ), λ) is the disjoint union of L-packets Π_{L}(G(ℝ), O^{∨}) (some of these may be empty). Specifying a single representation in an Lpacket Π_{L}(G(ℝ), O^{∨}) amounts to specifying a character of the two-group 𝕊_{ϕ} (ϕ ∈ O^{∨}), a problem we prefer to avoid.

On the other hand Π(G(ℝ), λ) is also the disjoint union of R-packets Π_{R}(G(ℝ), O, λ) (see (1.11)). Again specifying a single representation in an R-packet requires specifying a character of a two-group, in this case the component group of a torus.

The key to our parametrization is that the intersection of an L-packet and an R-packet is at most one element. We thereby obtain a classification in terms of pairs of packets, i.e. pairs of orbits. Here is a weak version of this result, which does not require any further assumptions on G(ℂ):

Lemma 1.16 ([27], Proposition 8.3) Suppose Π_{R}(G(ℝ), O, λ) is an Rpacket, and Π_{L}(G(ℝ), O^{∨}) is an L-packet. Then Π_{L}(G(ℝ), O^{∨})∩Π_{R}(G(ℝ), O, λ) has at most one element.

This reduces the problem of parametrizing Π(G(ℝ), λ) to the following problems. Let K(ℂ) be the complexification of a maximal compact subgroup of G(ℝ).

(1) Parametrize K(ℂ)-orbits on B,

(2) Parametrize G^{∨}(ℂ)-orbits on Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ).

It turns out that (2) is equivalent to

(2′) Parametrize K^{∨}(ℂ)-orbits on B^{∨}.

Here B^{∨} is the variety of Borel subgroups of G^{∨}(ℂ) and K^{∨}(ℂ) is the fixed points of an involution of G^{∨}(ℂ), i.e. the analogue of (1) on the dual side. We also need to determine when the intersection of an L-packet and an R-packet is non-empty.

We therefore turn to the problem of computing the space of K(ℂ) orbits on B, before returning the parametrization of Π(G(ℝ), λ) in Section 1.6.

1.5 Parametrization of K Orbits on the Flag Variety

Fix G(ℂ). As in Section 1.2 we are interested in computing the space of K(ℂ)-orbits on B, where K(ℂ) is the fixed points of a Cartan involution of G(ℂ). It turns out it is easier to treat all Cartan involutions simultaneously. For the purposes of this Overview we make two simplifying assumptions:

(1) the center of G(ℂ) is trivial,

(2) every automorphism of G(ℂ) is inner.

Assuming (1), condition (2) is equivalent to:

(2′) the Dynkin diagram of G(ℂ) has no nontrivial automorphisms.

For example this holds if G(ℂ) is a simple adjoint group of type B_{n},C_{n},E_{7}, E_{8},F_{4} or G_{2}.

Under assumption (2) every involutive automorphism of G(ℂ) is of the form int(x) for some involution x ∈ G(ℂ) (where int(x) is conjugation by x).

It follows that the (equivalence classes of) real forms of G(ℂ) are in bijection with conjugacy classes of involutions in G(ℂ). If x is such an involution then θ_{x} = int(x) is an involution of G(ℂ), and is the Cartan involution of a real form. Conversely the Cartan involution of every real form is of the form θ_{x}. See Section 3.

Definition 1.17 (Definition 9.2) Assuming (1) and (2), define

(1.18)  X (G(ℂ)) = {x ∈ Norm_{G(ℂ)}(H(ℂ)) | x^{2} = 1}/H(ℂ).

The quotient is by conjugation by H(ℂ). This is a finite set.

Let {x_{1}, …, x_{n}} be a set of representatives of the conjugacy classes of involutions of G(ℂ). For 1 ≤ i ≤ n let θ_{i} = int(x_{i}), K_{i}(ℂ) = G(ℂ)^{θ_{i}} and write G_{i}(ℝ) for the corresponding real form of G(ℂ). Fix a Cartan subgroup H(ℂ).

Proposition 1.19 Assume (1) and (2). There is a natural bijection

(1.20)  X (G(ℂ)) ⟷^{1−1} ∐_{i=1}^{n} K_{i}(ℂ)\B.

Sketch of proof. Let

(1.21) P = {(x, B(ℂ)) | x ∈ G(ℂ), x^{2} = 1, B(ℂ) a Borel subgroup}/G(ℂ).

Every element of P is conjugate to one of the form (x_{i}, B(ℂ)); the set of conjugacy classes of pairs (x_{i}, B(ℂ)) is isomorphic to K_{i}(ℂ)\B. This gives a

bijection P ⟷^{1−1} ∐_{i=1}^{n} K_{i}(ℂ)\B.

On the other hand fix a Borel subgroup B_{0}(ℂ) containing H(ℂ). Every element of P is conjugate to one of the form (x, B_{0}(ℂ)). Furthermore by conjugating by B_{0}(ℂ) we may assume x ∈ Norm_{G(ℂ)}(H(ℂ)), which gives a bijection P ⟷^{1−1} X (G(ℂ))). See Section 8. □

We turn next to the computation of Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ)/G^{∨}(ℂ). A remarkable fact, mentioned at the end of the previous section, is that the space X (applied on the dual side) provides this parametrization. To make this precise it is convenient to assume that the center of G^{∨}(ℂ) is trivial.

Fix a Cartan subgroup H^{∨}(ℂ) of G^{∨}(ℂ). Applying Definition 1.17 to G^{∨Γ} we have

(1.22)  X (G^{∨}(ℂ)) = {x ∈ Norm_{G^{∨}(ℂ)}(H^{∨}(ℂ)) | x^{2} = 1}/H^{∨}(ℂ).

Proposition 1.23 Assume G^{∨}(ℂ) is adjoint. There is a natural bijection

(1.24)  X (G^{∨}(ℂ)) ⟷^{1−1} Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ)/G^{∨}(ℂ).

Sketch of proof. We may view λ as an element of the Lie algebra of H^{∨}(ℂ). If x ∈ Norm_{G^{∨}(ℂ)}(H^{∨}(ℂ)), x^{2} = 1 define

(1.25)  \begin{aligned}ϕ(z) & = z^{λ}\overline{z}^{Ad(x)λ} \\ ϕ(j) & = e^{−πiλ}x. \end{aligned}

Then ϕ ∈ Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ), and every element of Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ) is conjugate to one of this form. See Section 7. □

Now assume Z(G(ℂ)) and Z(G^{∨}(ℂ)) are trivial, and Out(G(ℂ)) = 1. Then Proposition 1.19 holds here as well, so there are bijections

(1.26)  Hom_{adm}(W_{ℝ}, G^{∨Γ}, λ) ⟷^{1−1} X (G^{∨}(ℂ)) ⟷^{1−1} ∐_{i=1}^{m} K_{i}^{∨}(ℂ)\B^{∨}.

Here B^{∨} is the space of Borel subgroups of G^{∨}(ℂ), y_{1}, …, y_{m} are representatives of the conjugacy classes of involutions in G^{∨}(ℂ), and for each i K_{i}^{∨}(ℂ) = Cent_{G^{∨}(ℂ)}(y_{i}).

1.6 The parameter space Z

We now combine the results of the previous section with Lemma 1.16 to define the parameter space Z for Π(G(ℝ), λ).

To avoid some technical issues in the previous section we assumed the center of G(ℂ) is trivial and G(ℂ) has no outer automorphisms. For consideration of L(G^{∨Γ}, λ) we assumed the corresponding facts for G^{∨}(ℂ). Therefore we assume G(ℂ) is both simply connected and adjoint. We may as well also assume G(ℂ) is simple, i.e. of type G_{2}, F_{4} or E_{8}.

As in the previous section let x_{1}, …, x_{n} be representaitves of the conjugacy classes of involutions of G(ℂ), and let K_{i}(ℂ) = Cent_{G(ℂ)}(x_{i}). Dually let y_{1}, …, y_{m} be representatives of the conjugacy classes of involutions of G^{∨}(ℂ), and let K_{j}^{∨}(ℂ) = Cent_{G^{∨}(ℂ)}(y_{j}).

Definition 1.27 Assume G(ℂ) is simple, simply connected and adjoint. Fix Cartan subgroups H(ℂ) of G(ℂ) and H^{∨}(ℂ) of G^{∨}(ℂ). Let 𝔥, 𝔥^{∨} be the Lie algebras of H(ℂ) and H^{∨}(ℂ), respectively. Let

(1.28)(a) X = X (G(ℂ)) = {x ∈ Norm_{G(ℂ)}(H(ℂ)) | x^{2} = 1}/H(ℂ) X^{∨} = X (G^{∨}(ℂ)) = {y ∈ Norm_{G^{∨}(ℂ)}(H^{∨}(ℂ)) | y^{2} = 1}/H^{∨}(ℂ).

There is a natural adjoint map End(𝔥) ∋ X ↦ X^{t} ∈ End(𝔥^{∨}). Let

(1.29)  Z = {(x, y) ∈ X × X^{∨} | (Ad(x)|_{𝔥})^{t} = −Ad(y)|_{𝔥^{∨}}}.

By (1.20) and (1.26) Z may be viewed as a subset of

(1.30)  ∐_{i=1}^{n} K_{i}(ℂ)\B × ∐_{i=1}^{m} K_{i}^{∨}(ℂ)\B^{∨}.

See (10.7). Here is a special case of the main result:

Theorem 1.31 (Theorem 7.15) Assume G(ℂ) is simple, simply connected and adjoint. Write G_{1}(ℝ), …, G_{n}(ℝ) for the equivalence classes of real forms of G(ℂ).

There is a natural bijection

(1.32)  Z ⟷^{1−1} ∐_{i=1}^{n} Π(G_{i}(ℝ), λ).

This is a refinement of Lemma 1.16, and is a restatement of [1, Theorem 2-12]. Sketch of proof.

Here are three ways to think of the proof of this result.

Fix 1 ≤ i ≤ n and 1 ≤ j ≤ m. Also fix an orbit O of K_{i}(ℂ) on B, and an orbit O^{∨} of K_{j}^{∨}(ℂ) on B^{∨}. By Lemma 1.16 Π_{L}(G_{i}(ℝ), O^{∨}) ∩ Π_{R}(G_{i}(ℝ), O, λ) consists of at most element. Condition (1.29) makes this intersection nonempty, and we obtain every representation exactly once this way.

Alternatively, fix 1 ≤ i ≤ m and an orbit O^{∨} of K_{i}^{∨}(ℂ) on B^{∨}. Fix 1 ≤ j ≤ n, and let Π_{L}(G_{j}(ℝ), O^{∨}) be the corresponding L-packet. The choice of an orbit of K_{j}(ℂ) on B defines a character χ of 𝕊_{ϕ}, and by Proposition 1.14 this defines a representation of G_{j}(ℝ). This is the proof in Section 7 (see [1]).

Finally, fix 1 ≤ i ≤ n, an orbit O of K_{i}(ℂ) on B, and let Π_{R}(G_{i}(ℝ), O, λ) be the corresponding R-packet. The data needed to specify a D_{λ} module supported on O is precisely an orbit of K_{j}^{∨}(ℂ) on B^{∨} for some 1 ≤ j ≤ m. By Proposition 1.9 this defines an irreducible representation of G_{i}(ℝ). □

As is clear from the statement, to explicitly parametrize Π(G(ℝ), λ) the main issue is to understand the spaces X and X^{∨}. It is not immediately obvious how to explicitly compute these, but this can be done using the Tits group. This is discussed in Section 15. In any event all of the structural data discussed in Section 1 can be read off from the space X.

Proposition 1.33 (Proposition 12.19) Use the notation of Theorem 1.31.

(1) The real forms of G are parametrized by X ∩ H(ℂ)/W. Write x_{1}, …, x_{n} for representatives of this set.

(2) X ⟷^{1−1} ∐_{i=1}^{n} K_{x_{i}}(ℂ)\B.

(3) Associated to each x ∈ X is a pair (G_{x}(ℝ), H_{x}(ℝ)) consisting of a real form of G(ℂ) and a Cartan subgroup of G_{x}(ℝ). This induces a bijection between X /W and the union, over real forms G(ℝ) of G(ℂ), of the conjugacy classes of Cartan subgroups of G(ℝ).

(4) For x ∈ X we have

(1.34)  W (G_{x}(ℝ), H_{x}(ℝ)) ≃ Stab_{W} (x).

See Section 12.1.

It is clear from the discussion that the setting is entirely symmetric in G(ℂ) and G^{∨}(ℂ). This is a manifestation of Vogan duality [27], which is an essential guiding principal in the definitions. By symmetry we have, in the setting of (1.32),

(1.35)  ∐_{i=1}^{n} Π(G_{i}(ℝ), λ) ⟷^{1−1} Z ⟷^{1−1} ∐_{j=1}^{m} Π(G_{j}^{∨}(ℝ), λ^{∨})

(here λ^{∨} is a regular integral infinitesimal character for G^{∨}(ℂ)). See Corollary 10.8 for the general statement, without the restrictions on G(ℂ). In fact this is a refinement of Vogan duality of [27], which considers only a single real form at a time. See [3, Theorems 1.24 and 15.12].

This explains the nature of R-packets: it is clear from the discussion that the Vogan dual of an R-packet for G(ℝ) is an L-packet for some real form of G^{∨}(ℂ).

For ease of exposition in this Overview we have made various assumptions. Removing these constraints involves a number of closely related technical issues:

(1) If G(ℂ) is not adjoint we weaken the assumption x^{2} = 1 to x^{2} ∈ Z(G(ℂ)). This introduces the notion of strong real form (Section 5).

(2) If (the derived group of) G(ℂ) is not simply connected there are a finite number of different infinitesimal characters at which the representation theory looks different. Dually, since G^{∨}(ℂ) is not necessarily adjoint, we allow y^{2} ∈ Z(G^{∨}(ℂ)).

(3) If ρ does not exponentiate to G(ℂ) we allow characters of the ρ-cover of H(ℝ) [4, Definition 8.11].

(4) If G(ℂ) admits non-trivial outer automorphisms, we specify an involution γ in Out(G(ℂ)). Then all of the objects discussed above become “twisted” by γ.

Here is an example which takes some of these issues into account. We change notation to be more in agreement with the rest of the paper.

Let G = Sp(2n, ℂ) and let G^{∨} be the dual group, i.e. SO(2n + 1, ℂ). Fix Cartan subgroups H of G and H^{∨} of G^{∨}, with Lie algebras 𝔥 and 𝔥^{∨}.

Theorem 1.36 The irreducible representations of Sp(2n, ℝ) with the same infinitesimal character as the trivial representation are parametrized by:

{(x, y)}/H × H^{∨}

where

(1.37) x ∈ Norm_{G}(H), x^{2} = −I, y ∈ Norm_{G^{∨}}(H^{∨}) y^{2} = I,

and

(1.38)  (Ad(x)|_{𝔥})^{t} = −Ad(y)|_{𝔥^{∨}}.

The quotient is by the conjugation action of H × H^{∨}. This is a finite set. The number of elements is given by the following table:

n 1 2 3 4  5  6  7  8  9
4 18 88 460 2,544 1,4776 89,632 565,392 3,695,680

This set also parametrizes the irreducible representations of real forms SO(p, q) of SO(2n + 1, ℂ) with trivial infinitesimal character. This is an example of Vogan duality (see (1.35)).

If we instead require x^{2} = I this parametrizes representations of the groups Sp(p, q). Dually we obtain representations of SO(p, q) with infinitesimal character 2ρ.

We conclude this section with a comment about the requirements that the infinitesimal character λ be regular and integral. It is straightforward to remove both of these conditions, but introduces some extra complications.

First of all if λ is regular but not integral we replace G^{∨}(ℂ) with a subgroup whose root system is dual to the integral root system defined by λ. This follows the program of [27], and with this change many of the preceding constructions hold.

Secondly if λ is singular (and possibly non-integral) let λ′ be a regular element such that λ − λ′ is a sum of roots. By Zuckerman’s translation principle [28] Π(G(ℝ), λ) is a an explicitly computable subset of Π(G(ℝ), λ′), thereby reducing this case to the one of regular infinitesimal character.

1.7 Outline

Here is an outline of the contents of the paper.

Section 2 defines the basic objects of study, i.e. reductive groups and root data. In Section 3 we discuss real forms of a complex group.

We put the information from Sections 2 and 3 together to define basic data in Section 4. This consists of either a pair (G, γ) consisting of a complex reductive group and an involution in Out(G), or a pair (D_{b}, γ) consisting of a based root datum and an involution of it.

Given basic data we define the extended group G^{Γ} = G ⋊ Γ in Section 5. We also define the notions of strong involution and strong real form. Harish-Chandra modules for strong real forms are discussed in Section 6.

The first main step in constructing the parameter space is L-data (Section 7). The relation with K-orbits on G/B is the subject of Section 8.

The primary combinatorial construction is the one-sided parameter space X of Section 9. Once we have X it is straightforward to define the parameter space Z ⊂ X × X^{∨}. The main result is Theorem 10.3.

After stating the main result, we go back down into some of the details of the space X in Sections 11-14 and relate this space to structure theory of G. A summary of the relationship between X and structure theory of G is found in Section 12.1. We discuss the reduced parameter space X^{r} in Section 13. In Section 15 we use the Tits group to explicitly compute the space X.

Some examples are discussed in the body of the paper. In particular the very informative cases of SL(2) and P SL(2) are discussed in examples 3.4, 5.6, 12.20, and 12.25. More examples may be found in [2].

2 Reductive Groups and Root Data

For many purposes one may identify a connected reductive algebraic group with its group of complex points. For the discussion of real forms (Section 3), and to keep the exposition as elementary as possible, we choose to work with complex groups. Experts, and those with an interest in other fields, may wish to convert to the language of algebraic groups where appropriate.

We now describe the parameters for a connected reductive complex group. These are provided by root data and based root data. Good references are the books by Humphreys [11] and Springer [23].

We begin with a pair X, X^{∨} of free abelian groups of finite rank, together with a perfect pairing ⟨ , ⟩ : X × X^{∨} → ℤ. Suppose Δ ⊂ X, Δ^{∨} ⊂ X^{∨} are finite sets, equipped with a bijection α → α^{∨}. For α ∈ Δ define the reflection s_{α} ∈ Hom(X, X):

s_{α}(x) = x − ⟨x, α^{∨}⟩α (x ∈ X)

and define s_{α^{∨}} ∈ Hom(X^{∨}, X^{∨}) similarly.

A root datum is a quadruple

(2.1)  D = (X, Δ, X^{∨}, Δ^{∨})

where X, X^{∨} are free abelian groups of finite rank, in duality via a perfect pairing ⟨ , ⟩, and Δ, Δ^{∨} are finite subsets of X, X^{∨}, respectively. We assume there is a bijection Δ ∋ α ↦ α^{∨} ∈ Δ^{∨} such that for all α ∈ Δ,

(2.2)  ⟨α, α^{∨}⟩ = 2, s_{α}(Δ) = Δ, s_{α^{∨}}(Δ^{∨}) = Δ^{∨}.

Suppose we are given X, X^{∨} and finite subsets Δ ⊂ X and Δ^{∨} ⊂ X^{∨}. By [7, Lemma VI.1.1] applied to ℚ⟨Δ⟩ and ℚ⟨Δ^{∨}⟩ there is at most one bijection α ↦ α^{∨} satisfying (2.2). Alternatively suppose we are given only a finite subset Δ of X, satisfying X ⊂ ℚ⟨Δ⟩. By (loc. cit.) there is at most one subset Δ^{∨}, and bijection α ↦ α^{∨}, satisfying (2.2). The condition X ⊂ ℚ⟨Δ⟩ holds if and only if the corresponding group is semisimple.

Suppose D_{i} = (X_{i}, Δ_{i}, X_{i}^{∨}, Δ_{i}^{∨}) (i = 1, 2) are root data. We say they are isomorphic if there is an isomorphism ϕ ∈ Hom(X_{1}, X_{2}) satisfying ϕ(Δ_{1}) = Δ_{2} and ϕ^{t}(Δ_{2}^{∨}) = Δ_{1}^{∨}. Here ϕ^{t} ∈ Hom(X_{2}^{∨}, X_{1}^{∨}) is defined by

(2.3)  ⟨ϕ(x_{1}), x_{2}^{∨}⟩_{2} = ⟨x_{1}, ϕ^{t}(x_{2}^{∨})⟩_{1} (x_{1} ∈ X_{1}, x_{2}^{∨} ∈ X_{2}^{∨}).

Let G be a connected reductive complex group and choose a Cartan subgroup H of G. Let X^{∗}(H), X_{∗}(H) be the character and co-character lattices of H respectively. We have canonical isomorphisms

(2.4)  𝔥 ≃ X_{∗}(H) ⊗_{ℤ} ℂ, 𝔥^{∗} ≃ X^{∗}(H) ⊗_{ℤ} ℂ

where 𝔥 is the Lie algebra of H and 𝔥^{∗} = Hom(𝔥, ℂ). (If G has rank n then H ≃ (ℂ^{×})^{n}, X_{∗}(H) ≃ ℤ^{n} and 𝔥 ≃ ℂ^{n}.) Let Δ = Δ(G, H) be the set of roots of H in G, and Δ^{∨} = Δ^{∨}(G, H) the corrresponding co-roots. Associated to (G, H) is the root datum

(2.5)  D(G, H) = (X^{∗}(H), Δ, X_{∗}(H), Δ^{∨}).

If H′ is another Cartan subgroup then there is an element g ∈ G so that gHg^{−1} = H′. Let D′ = (X^{∗}(H′), Δ(G, H′), X_{∗}(H′), Δ^{∨}(G, H′)) be the corresponding root datum. The inverse transpose action on characters induces an isomorphism

(Ad(g)^{t})^{−1} : X^{∗}(H) → X^{∗}(H′)

which gives an isomorphism D(G, H) ≃ D(G, H′).

Now suppose in addition to H we have chosen a Borel subgroup B containing H. Let Δ^{+} be the corresponding set of positive roots of Δ, with simple roots Π. Then Π^{∨} = {α^{∨} | α ∈ Π} is a set of simple roots of Δ^{∨}, and

(2.6)  D_{b}(G, B, H) = (X, Π, X^{∨}, Π^{∨})

is a based root datum. If H′ ⊂ B′ are another Cartan and Borel subgroup then there is a canonical isomorphism D_{b}(G, B, H) ≃ D_{b}(G, B′, H′).

Each root datum is the root datum of a reductive algebraic group, which is determined uniquely up to isomorphism, and the same holds for based root data.

Note that a connected reductive complex group G of rank n is determined by a small finite set of data: two sets (of order the semisimple rank of G) of integral n-vectors, subject only to condition (2.2), which may be expressed by requiring that the matrix of dot products is a Cartan matrix.

Example 2.7 Suppose G is of rank 2 and semisimple rank 1. Then a root datum is given by an ordered pair (v, w) with v, w ∈ ℤ^{2}, satisfying v · w = 2. Equivalence is given by the action of GL(2, ℤ), where g ·(v, w) = (gv,^{t}g^{−1}w). It is an interesting exercise to see that there are precisely three such pairs, up to equivalence: ((2, 0), (1, 0)), ((1, 0), (2, 0)) and ((1, 1), (1, 1)), corresponding to SL(2, ℂ) × ℂ^{×}, P GL(2, ℂ) × ℂ^{×} and GL(2, ℂ), respectively.

If D = (X, Δ, X^{∨}, Δ^{∨}) is a root datum then the dual root datum is defined to be D^{∨} = (X^{∨}, Δ^{∨}, X, Δ). Given G with root datum D = (X, Δ, X^{∨}, Δ^{∨}) the dual group is the group G^{∨} defined by D^{∨}. We define duality of based root data similarly.

2.1 Automorphisms

There is an exact sequence

(2.8)  1 → Int(G) → Aut(G) → Out(G) → 1

where Int(G) is the group of inner automorphisms of G, Aut(G) is the group of (holomorphic) automorphisms of G, and Out(G) ≃ Aut(G)/Int(G) is the group of outer automorphisms.

A splitting datum or pinning for G is a set (B, H, {X_{α}}) where B is a Borel subgroup, H is a Cartan subgroup contained in B and {X_{α}} is a set of root vectors for the simple roots of H in B.

Definition 2.9 An involution of G is said to be distinguished if it preserves a splitting datum.

For example an inner involution is distinguished if and only if it is the identity; this is the Cartan involution of the compact real form. More generally, among all of the involutions mapping to a fixed involution in Out(G), a distinguished involution makes as many roots as possible imaginary and compact (cf. Section 12); it is the Cartan involution of a “maximally compact” real form of G in the given inner class (cf. Definition 3.6).

The group Int(G) acts simply transitively on the set of splitting data. Given a splitting datum (B, H, {X_{α}}) this gives an isomorphism

(2.10)(a)  ϕ_{S} : Out(G) ≃ Stab_{Aut(G)}(S) ⊂ Aut(G)

and this is a splitting of the exact sequence (2.8). We call a splitting distinguished if it fixes a splitting datum. We obtain isomorphisms

(2.10)(b)  Out(G) ≃ Aut(D_{b}) ≃ Aut(D)/W.

If G is semisimple then there is an injection of Out(G) into the automorphism group of the Dynkin diagram of G; if G is semisimple and simply connected or adjoint then this is an isomorphism.

If τ ∈ Aut(D) then −τ^{t} ∈ Aut(D^{∨}) (cf. 2.3). Now suppose τ ∈ Aut(D_{b}). While −τ^{t} is probably not in Aut(D_{b}^{∨}), if we let w_{0} be the long element of the Weyl group we have −w_{0}τ^{t} ∈ Aut(D_{b}^{∨}).

Definition 2.11 Suppose τ ∈ Aut(D_{b}). Let τ^{∨} = −w_{0}τ^{t} ∈ Aut(D_{b}^{∨}). This defines a bijection Aut(D_{b}) ⟷^{1−1} Aut(D_{b}^{∨}). By (2.10)(b) we obtain a bijection Out(G) ↔ Out(G^{∨}) by composition:

(2.12)  Out(G) ↔ Aut(D_{b}) ↔ Aut(D_{b}^{∨}) ↔ Out(G^{∨}).

For γ ∈ Out(G) we write γ^{∨} for the corresponding element of Out(G^{∨}). The map γ ↦ γ^{∨} is a bijection of sets.

Remark 2.13 This is not necessarily an isomorphism of groups. For example the identity goes to the image of −w_{0} in Out(G), which is the identity if and only G is semisimple and −1 ∈ W.

Example 2.14 Let G = P GL(n) (n ≥ 3). Then G^{∨} = SL(n) and Out(G) ≃ Out(G^{∨}) ≃ ℤ/2ℤ. If γ = 1 ∈ Out(G) then γ^{∨} is the non-trivial element of Out(G^{∨}). It is represented by the automorphism τ^{∨}(g) = ^{t}g^{−1} of G^{∨}.

3 Involutions of Reductive Groups

Fix a connected reductive complex group G. A real form of G is a subgroup G(ℝ) which is the fixed points of an antiholomorphic involution of G. Two such real forms are said to be equivalent if they are conjugate by G. The Cartan involution provides a description of real forms in terms of holomorphic involutions which is better suited to our purposes. We illustrate this with an example.

Example 3.1 Let G = GL(n, ℂ). The group GL(n, ℝ) is a real form of GL(n, ℂ); it is the fixed points of the antiholomorphic involution σ(g) = \overline{g}. The orthogonal group O(n) is a maximal compact subgroup of GL(n, ℝ); it is the fixed points of the involution θ(g) = ^{t}g^{−1} of GL(n, ℝ). This involution extends to a holomorphic involution of GL(n, ℂ), with fixed points O(n, ℂ).

Suppose σ is an antiholomorphic involution of G, with fixed points G(ℝ). Then there is a holomorphic involution θ of G such that G(ℝ)^{θ} is a maximal compact subgroup of G(ℝ). It turns out this gives a bijection on the level of G-conjugacy classes. See [19, Theorem 5.1.4], [4] and [10, Chapter X, Section 1].

Theorem 3.2 The map σ ↦ θ gives a bijection between G-conjugacy classes of antiholomorphic involutions of G and G-conjugay classes of holomorphic involutions of G.

Using this result we classify real forms in terms of holomorphic involutions. We also prefer to incorporate the notion of equivalence in the definition of real forms.

Definition 3.3 Let G be a connected reductive complex group. An involution of G is an involution in Aut(G), i.e. a holomorphic automorphism θ of G satisfying θ^{2} = 1. A real form of G is a G-conjugacy class of involutions.

Our definition of equivalence of real forms differs from the usual one in one subtle way: it only allows conjugacy by G, rather than all of Aut(G). The theorem also holds with G-conjugacy replaced by conjugacy by Aut(G), and this is how it is usually stated (for example [19, Theorem 5.1.4]).

Example 3.4 We consider the case of SL(2, ℂ). Up to conjugacy there are two antiholomorphic involutions of SL(2, ℂ): σ^{s}(g) = \overline{g} and σ^{c}(g) = ^{t}\overline{g}^{−1}.

These are the two real forms G(ℝ) = SL(2, ℝ) (split) and G(ℝ) = SU(2) (compact) of SL(2, ℂ), respectively.

Equivalently SL(2, ℂ) has two equivalence classes of holomorphic involutions. Let θ^{s}(g) = tgt^{−1} where t = diag(i, −i). Then K(ℂ) = G^{θ^{s}} = ℂ^{×}, so K(ℝ) = S^{1}, the maximal compact subgroup of the corresponding real form SL(2, ℝ). On the other hand let θ^{c}(g) = g, so K(ℂ) = SL(2, ℂ), K(ℝ) = SU(2), and the corresponding real form is SU(2).

Remark 3.5 By Theorem 3.2 there is a bijection between equivalence classes of real forms in the usual sense of G(ℝ) and antiholomorphic involutions, and real forms in the sense of Definition 3.3. We work almost entirely with the latter notion; in the few places we refer to the former the distinction will be clear since we will refer to a real form G(ℝ).

Definition 3.6 An involution θ ∈ Aut(G) (Definition 3.3) is in the inner class of γ ∈ Out(G) if θ maps to γ in the exact sequence (2.8). If θ, θ′ are involutions of G we say θ is inner to θ′ if θ and θ′ have the same image in Out(G).

This corresponds to the usual notion of inner form [23, 12.3.7].

Remark 3.7 The results of [3] are stated in terms of antiholomorphic, rather than holomorphic, involutions. See Remark 5.18.

Remark 3.8 As discussed at the beginning of Section 2, this is the one situation in which we need complex, as opposed to algebraic, groups. Once we are in the setting of Cartan involutions, we could replace G with the underlying algebraic group 𝔾, and holomorphic involutions of G with algebraic involutions of 𝔾. This is the setting of [21].

4 Basic Data

The setting for our mathematical questions will be:

(1) A connected reductive complex group G,

(2) An involution γ ∈ Out(G).

We refer to (G, γ) as basic data. We say (G, γ) is equivalent to (G′, γ′) if there is an isomorphism ϕ : G → G′ such that γ′ ∘ ϕ = ϕ ∘ γ.

On the other hand the software works entirely in the setting of based root data. Given (G, γ), let (B, H, {X_{α}}) be a splitting datum (Section 2.1). From these we obtain:

(a) A based root datum D_{b} = D_{b}(G, B, H),

(b) An involution γ of D_{b} (cf. (2.10)(b)).

The based root datum D_{b} and its involution are independent of the choice of splitting datum, up to canonical isomorphism. Conversely, given a based root datum D_{b} with an involution γ we can construct (G, γ), uniquely determined up to isomorphism.

Given basic data (G, γ), choose a splitting datum (B, H, {X_{α}}). A key to our algorithm is that (B, H, {X_{α}}) is fixed once and for all. This enables us to do all of our constructions on a fixed Cartan subgroup. Let W = W (G, H) be the Weyl group.

The weight lattice for G is

(4.1)  P = {λ ∈ X^{∗}(H) ⊗ ℂ | ⟨λ, α^{∨}⟩ ∈ ℤ for all α ∈ Δ}

and dually the co-weight lattice for G is

(4.2)(a)  P^{∨} = {λ^{∨} ∈ X_{∗}(H) ⊗ ℂ | ⟨α, λ^{∨}⟩ ∈ ℤ for all α ∈ Δ}.

These are actually lattices only if G is semisimple. We may identify 2πiX_{∗}(H) with the kernel of exp : 𝔥 → H(ℂ). Under this identification

(4.2)(b)  P^{∨} = {λ^{∨} ∈ 𝔥 | exp(2πiλ^{∨}) ∈ Z(G)}.

We write P (G, H) and P^{∨}(G, H) to indicate the dependence on G and H. We also define

(4.2)(c)  P_{reg} = {λ ∈ P | ⟨λ, α^{∨}⟩ ≠ 0 for all α ∈ Δ}

and

(4.2)(d)  P_{reg}^{∨} = {λ^{∨} ∈ P^{∨} | ⟨α, λ^{∨}⟩ ≠ 0 for all α ∈ Δ}.

By duality we obtain the dual based root datum D_{b}^{∨}, an involution γ^{∨} of D_{b}^{∨} (cf. Definition 2.11), the dual group G^{∨}, and thus dual basic data (G^{∨}, γ^{∨}). In particular G^{∨} comes equipped with fixed Cartan and Borel subgroups H^{∨} ⊂ B^{∨}. We have X^{∗}(H) = X = X_{∗}(H^{∨}) and X_{∗}(H) = X^{∨} = X^{∗}(H^{∨}). These are canonical identifications. By (2.4) applied to H and H^{∨} these induce identifications 𝔥 = 𝔥^{∨∗} and 𝔥^{∗} = 𝔥^{∨}. Note that P (G, H) = P^{∨}(G^{∨}, H^{∨}), and by (4.2)(b) we have

(4.3)  P (G, H) = {λ ∈ 𝔥^{∨} | exp(2πiλ) ∈ Z(G^{∨})}.

We are also given the bijection Δ = Δ(G, H) ∋ α → α^{∨} ∈ Δ^{∨} = Δ(G^{∨}, H^{∨}). We identify W (G, H) with W (G^{∨}, H^{∨}) by the map W (G, H) ∋ w → w^{t} ∈ W (G^{∨}, H^{∨}) (cf. (2.3)); equivalently, s_{α} → s_{α^{∨}}.

5 Extended Groups and Strong Real Forms

Fix basic data (G, γ) as in Section 4, and a splitting datum (H, B, {X_{α}}) (cf. Section 2.1). This gives us a distinguished splitting of the exact sequence (2.8), taking γ to a distinguished involution of G (Definition 2.9) which we also denote γ. Let Γ = {1, σ} be the Galois group of ℂ/ℝ.

Definition 5.1 The extended group for (G, γ) is the semi-direct product

(5.2)  G^{Γ} = G ⋊ Γ

where σ ∈ Γ acts on G by the distinguished involution γ.

We say the element 1 × σ is distinguished, and denote it δ.

Recall γ is a Cartan involution of a maximally compact form in this inner class.

Example 5.3 Suppose γ = 1, so G^{Γ} = G × Γ. Note that γ is the Cartan involution of the compact real form of G, and this is the “equal rank” case. Every real form G(ℝ) in this inner class contains a compact Cartan subgroup, and every involution in this inner class is contained in Int(G).

In particular suppose G is semisimple and the Dynkin diagram of G has no non-trivial automorphisms. Then γ is necessarily trivial, and every involution of G is inner.

Example 5.4 Suppose 1 ≠ γ ∈ Out(G). The most convenient way to compute the corresponding distinguished involution γ of G (cf. Definition 2.9) is to list the real forms in this inner class, and choose the most compact one.

For example let G = SL(n, ℂ) (n ≥ 3). There is a unique outer automorphism of the Dynkin diagram, corresponding to the non-trivial element γ ∈ Out(G). If n is odd the only real form in this inner class is the split group SL(n, ℝ), and we take γ(g) = ^{t}g^{−1}, the Cartan involution of SL(n, ℝ).

If n is even then G has two real forms, SL(n, ℝ) and SL(n/2, ℍ) where ℍ is the quaternion algebra. We let γ act by a Cartan involution of SL(n/2, ℍ).

The extended group G^{Γ} encapsulates all of the real forms of G in the inner class defined by γ. That is if ξ ∈ G^{Γ}\G satisfies ξ^{2} ∈ Z(G) then int(ξ) is an involution in the inner class of γ. Conversely, if θ is in the inner class of γ, then there is an element ξ ∈ G^{Γ}\G with ξ^{2} ∈ Z(G) and θ = int(ξ). For the algorithm it is important to keep track of ξ, and not just θ = int(ξ).

Definition 5.5 A strong involution of G in the inner class of γ is an element ξ ∈ G^{Γ}\G such that ξ^{2} ∈ Z(G). The set of such strong involutions is denoted I(G, γ). We define a strong real form of G in the inner class of γ to be a G-conjugacy class of strong involutions.

For ξ ∈ I(G, γ) let θ_{ξ} = int(ξ) and K_{ξ} = Stab_{G}(ξ) = G^{θ_{ξ}}.

If γ is understood we refer to strong involutions and strong real forms of G.

Example 5.6 Recall (Example 3.4) SL(2, ℂ) has two real forms, θ^{c} = 1 (i.e. SU(2)) and θ^{s} = int(diag(i, −i)) (i.e. SL(2, ℝ) = SU(1, 1)). However SL(2, ℂ) has three strong real forms, i.e. conjugacy classes of strong involutions: ξ = I, −I and diag(i, −i). (Here and elsewhere, when γ = 1, we write ξ for the element (ξ, σ) ∈ G^{Γ}\G.) Then θ_{ξ} = θ^{c} if ξ = ±I, or θ_{ξ} = θ^{s} if ξ = diag(i, −i). We can think of these strong real forms as SU(2, 0), SU(0, 2) and SU(1, 1) ≃ SL(2, ℝ), respectively.

Now consider P SL(2, ℂ) ≃ SO(3, ℂ). This has two real forms: the compact group SO(3, ℝ) (with K(ℝ) = SO(3, ℝ), K(ℂ) = SO(3, ℂ)) and the split one SO(2, 1) (with K(ℝ) = O(2), K(ℂ) = O(2, ℂ)). Up to conjugacy there are two strong involutions: I and diag(−1, −1, 1).

The next Lemma is immediate from the definitions (cf. Definition 3.6).

Lemma 5.7 We have

I(G, γ)/G = {strong real forms in the inner class of γ}.

The map I(G, γ) ∋ ξ ↦ θ_{ξ} is a surjection from I(G, γ) to the set of involutions in the inner class of γ. It factors to a surjection

I(G, γ)/G ↠ {real forms of G in the inner class of γ}.

If G is adjoint this is a bijection.

Example 5.8 In Example 5.6 the map from strong real forms to real forms is bijective for P SL(2). If G = SL(2) the fiber of the map is a single element for the split real form SU(1, 1), and two elements (SU(2, 0) and SU(0, 2), so to speak) for the compact real form SU(2).

Example 5.9 This is generalization of Example 5.6. Suppose γ is the identity. The strong real forms of G are parametrized by

(5.10)(a)  {ξ ∈ G | ξ^{2} ∈ Z(G)}/G.

(As in Example 5.6 since γ = 1 we write ξ instead of (ξ, σ).) Every strong involution ξ is conjugate to an element of H so this set is the same as

(5.10)(b)  \{ξ ∈ H | ξ^{2} ∈ Z(G)\}/W = (\frac{1}{2}P^{∨}/X_{∗}(H))/W

where P^{∨} is the coweight lattice (cf. (4.2)(a)). In particular if G is adjoint the real forms of G are parametrized by

(5.10)(c)  (\frac{1}{2}P^{∨}/P^{∨})/W.

Example 5.11 Continuing with the preceding example, let G = P Sp(2n). In the usual coordinates P^{∨} ≃ ℤ^{n} ∪ (ℤ + \frac{1}{2})^{n}, and for representatives of (\frac{1}{2}P^{∨}/P^{∨})/W we may take

(5.12)  \frac{1}{4}(1, …, 1)

corresponding to P Sp(2n, ℝ) (the split group) and

(5.13)  \frac{1}{2}(\overbrace{1, …, 1}^{p}, \overbrace{0, …, 0}^{q}) (p ≤ [n/2])

corresponding to P Sp(p, q).

For G = Sp(2n) we have X_{∗}(H) = R^{∨} ≃ ℤ^{n}, and the strong real forms are parametrized by

(5.14)  (\frac{1}{2}[ℤ^{n} ∪ (ℤ + \frac{1}{2})^{n}]/ℤ^{n})/W.

For representatives we may take \frac{1}{4}(1, …, 1) and \frac{1}{2}(\overbrace{1, …, 1}^{p}, \overbrace{0, …, 0}^{q}) as before, where now 0 ≤ p ≤ n. Thus there are two strong real forms mapping to each real form Sp(p, q) for p ≠ q.

See [1, Example 1-17].

We will make frequent use of the following construction. Choose a set of representatives {ξ_{i} | i ∈ I} of the set of strong real forms. That is

(5.15)(a)  {ξ_{i} | i ∈ I} ≃ I(G, γ)/G.

If G is semisimple (in fact if Z(G)^{Γ} is finite, cf. (11.2)(3)) this is a finite set. For i ∈ I let

(5.15)(b)  θ_{i} = int(ξ_{i}), K_{i} = G^{θ_{i}}

and

(5.15)(c)  I_{i} = {ξ ∈ I(G, γ) | ξ is G-conjugate to ξ_{i}}.

The stabilizer of ξ_{i} in G is K_{i}, so I_{i} ≃ G/K_{i}, and we have

(5.15)(d)  I(G, γ) ≃ ∐_{i∈I} G/K_{i}.

Recall the distinguished involution θ_{δ} = int(δ) is the Cartan involution of a maximally compact real form in the inner class of γ. It is helpful to also invoke the most split form in this inner class. We say an involution of G is quasisplit if it is the Cartan involution of a quasisplit real form of G. For a characterization of quasisplit involutions see [4, Proposition 6.24] (where they are called principal). By [4, Theorem 6.14] there is a unique conjugacy class of quasisplit involutions in each inner class.

We emphasize the symmetry of the situation by summarizing this:

Lemma 5.16 Let G^{Γ} be the extended group for (G, γ).

(1) There exists a strong involution ξ ∈ G^{Γ} so that θ_{ξ} is distinguished. The involution θ_{ξ} is unique up to conjugation by G.

(2) There exists a strong involution η so that θ_{η} is quasisplit. The involution θ_{η} is unique up to conjugation by G.

Remark 5.17 The extended group G^{Γ} in [4, Definition 9.6] is defined in terms of a quasisplit involution, rather than a distinguished one. The equivalence of the two definitions is the content of [4, 9.7]. This discussion also shows that, applied to (G^{∨}, γ^{∨}), the group G^{∨Γ} is isomorphic to the L-group of the real forms of G in the iner class of γ.

Remark 5.18 Since in [3] we work with antiholomorphic involutions instead of holomorphic ones, (cf. Remark 3.7), the extended group G^{Γ} in [3, Chapter 3] is defined in terms of an antiholomorphic involution. The results are equivalent, but some translation is necessary between the two pictures.

6 Harish-Chandra modules for strong real forms

Fix basic data (G, γ) as in Section 2 and let I = I(G, γ) be the corresponding set of strong involutions. For ξ ∈ I let K_{ξ} = Stab_{G}(ξ) as usual, and define (𝔤, K_{ξ}) modules as in [26].

Definition 6.1 A Harish-Chandra module for a strong involution is a pair (ξ, π) where ξ ∈ I and π is a (𝔤, K_{ξ})-module. A Harish-Chandra module for a strong real form of G is the G-orbit of a pair (ξ, π), where g · (ξ, π) = (gξg^{−1}, π^{g}).

An infinitesimal character for 𝔤 may be identified, via the Harish-Chandra homomorphism with an orbit of W on 𝔥^{∗}. For λ ∈ 𝔥^{∗} we write χ_{λ} for the corresponding infinitesimal character; note that χ_{λ} = χ_{wλ} for all w ∈ W. If π is a Harish-Chandra module with infinitesimal character χ_{λ} we may simply say π has infinitesimal character λ.

Definition 6.2 We say λ and χ_{λ} are integral if λ ∈ P, and regular and integral if λ ∈ P_{reg} (cf. (4.2)(c)).

Definition 6.3 Given ξ ∈ I let Π(G, ξ) be the set of equivalence classes of irreducible (𝔤, K_{ξ})-modules with regular integral infinitesimal character. Let

(6.4)  Π(G, γ) = {(ξ, π) | ξ ∈ I, π ∈ Π(G, ξ)}/G.

If Λ is a subset of P_{reg} let Π(G, γ, Λ) ⊂ Π(G, γ) be the set of (equivalence classes of) pairs (ξ, π) for which the infinitesimal character of π is an element of Λ.

If we fix a set of representatives I of I/G as in (5.15) we have

(6.5)  Π(G, γ) ≃ ∐_{i∈I} Π(G, ξ_{i}).

Thus Π(G, γ) parametrizes Harish-Chandra modules for strong real forms of (G, γ), with regular integral infinitesimal character. With the obvious notation we also have Π(G, γ, Λ) = ∐_{i∈I} Π(G, ξ_{i}, Λ).

Fix ξ ∈ I. As a consequence of [16], associated to each G^{∨}-conjugacy class of admissible homomorphisms ϕ : W_{ℝ} → G^{∨Γ} is an L-packet Π_{ϕ}(G, ξ) (see Section 7). This is a finite set of (equivalence classes of) (𝔤, K_{ξ}) modules, all having the same infinitesimal character. Each L-packet is finite, and non-empty if ξ is quasisplit. The non-empty L-packets partition the set of irreducible Harish-Chandra modules for the strong involution ξ.

We define the large L-packet of ϕ to be the union, over all strong involutions, of L-packets, modulo our notion of equivalence:

(6.6)(a)  Π_{ϕ}(G, γ) = {(ξ, π) | ξ ∈ I, π ∈ Π_{ϕ}(G, ξ)}/G.

With I as in (5.15) we have

(6.6)(b)  Π_{ϕ}(G, γ) ≃ ∐_{i∈I} Π_{ϕ}(G, ξ_{i})

and each set Π_{ϕ}(G, ξ_{i}) is finite.

7 L-data

Fix basic data (G, γ), with corresponding dual data (G^{∨}, γ^{∨}) (cf. Section 4). Let G^{∨Γ} be the corresponding extended group (Definition 5.1). Recall (Remark 5.17) that G^{∨Γ} is isomorphic to the L-group of G. We begin by parametrizing admissible homomorphisms of the Weil group into G^{∨Γ} [6].

Let W_{ℝ} be the Weil group of ℝ. That is W_{ℝ} = ⟨ℂ^{×}, j⟩ where jzj^{−1} = \overline{z} and j^{2} = −1. An admissible homomorphism ϕ : W_{ℝ} → G^{∨Γ} is a continuous homomorphism such that ϕ(ℂ^{×}) consists of semisimple elements and ϕ(j) ∈ G^{∨Γ}\G^{∨}.

Suppose ϕ : W_{ℝ} → G^{∨Γ} is an admissible homomorphism. Then ϕ(ℂ^{×}) is contained in a Cartan subgroup H_{1}^{∨}(ℂ), and ϕ(e^{z}) = exp(2πi(zλ + \overline{z}ν)) for some λ, ν ∈ 𝔥_{1}^{∨}. Choose an inner isomorphism 𝔥_{1}^{∨} ≃ 𝔥^{∨}. As explained in Section 4 we have 𝔥^{∨} = 𝔥^{∗}, so we may identify λ with an element (which we still call λ) of 𝔥^{∗}, whose W orbit is well-defined. We define the infinitesimal character of ϕ to be the W -orbit of λ ∈ 𝔥^{∗}.

Recall P (G, H) = {λ ∈ 𝔥^{∗} | exp(2πiλ) ∈ Z(G^{∨})} (cf. (4.3)).

Definition 7.1 A complete one-sided L-datum for (G^{∨}, γ^{∨}) is a triple (η, B_{1}^{∨}, λ) where η is a strong involution of G^{∨} (Definition 5.5), B_{1}^{∨} is a Borel subgroup of G^{∨} and λ ∈ P (G, H) satisfies exp(2πiλ) = η^{2}. The group G^{∨} acts by conjugation on this data, and let

(7.2)  P_{c}(G^{∨}, γ^{∨}) = {complete one-sided L-data}/G^{∨}.

Fix a complete one-sided L-datum S_{c}^{∨} = (η, B_{1}^{∨}, λ). By [17] (also see [3, Lemma 6.18]) there is a θ_{η}-stable Cartan subgroup H_{1}^{∨} of B_{1}^{∨}, unique up to conjugacy by K_{η}^{∨} ∩ B_{1}^{∨}. Choose g ∈ G^{∨} such that gH^{∨}g^{−1} = H_{1}^{∨} and ⟨Ad(g)λ, α^{∨}⟩ ≥ 0 for all α ∈ Δ(B_{1}^{∨}, H_{1}^{∨}). Let λ_{1} = Ad(g)λ ∈ 𝔥_{1}^{∨}. Define ϕ : W_{ℝ} → G^{∨Γ} by:

(7.3)  \begin{aligned}ϕ(z) & = z^{λ_{1}} \overline{z}^{ηλ_{1}} (z ∈ ℂ^{×}) \\ ϕ(j) & = exp(−πiλ_{1})η. \end{aligned}

The first statement is shorthand for ϕ(e^{z}) = exp(zλ_{1} + Ad(η)\overline{z}λ_{1}). It is easy to see ϕ is an admissible homomorphism, and the G^{∨} conjugacy class of ϕ is independent of the choices of H_{1} and g. The next result follows readily.

Proposition 7.4 The map taking (η, B_{1}^{∨}, λ) to ϕ defined by (7.3) induces a bijection between P_{c}(G^{∨}, γ^{∨}) and the set of G^{∨}-conjugacy classes of admissible homomorphisms W_{ℝ} → G^{∨Γ}.

Recall (cf. 6.6) associated to an admissible homomorphism ϕ is a large L-packet Π_{ϕ}(G, γ). For S_{c}^{∨} ∈ P_{c}(G^{∨}, γ^{∨}) define ϕ by (7.3) and let

(7.5)  Π_{S_{c}^{∨}}(G, γ) = Π_{ϕ}(G, γ).

Fix (η, B_{1}^{∨}). If λ ∈ P_{reg}, S_{c}^{∨} = (η, B_{1}^{∨}, λ) is a complete one-sided Ldatum if and only if exp(2πiλ) = η^{2}. For λ satisfying this condition we may define the large L-packet Π_{S_{c}^{∨}}(G, γ). Any two such L-packets are related by a translation functor [28], and in this sense the choice of λ is not important. We therefore drop the parameter λ from complete L-data:

Definition 7.6 A one-sided L-datum for (G^{∨}, γ^{∨}) is a pair S^{∨} = (η, B_{1}^{∨}) where η is a strong involution of G^{∨} (Definition 5.5) and B_{1}^{∨} is a Borel subgroup of G^{∨}. Let

(7.7)  P(G^{∨}, γ^{∨}) = {one-sided L-data}/G^{∨}.

There is a well defined map

(7.8)  P(G^{∨}, γ^{∨}) ↦ Z(G^{∨})

taking (the equivalence class of) (η, B_{1}^{∨}) to η^{2}.

For the purposes of Proposition 7.4 we have defined one-sided L-data P(G^{∨}, γ^{∨}) for (G^{∨}, γ^{∨}). It is evident that the definition is symmetric, and applies equally to (G, γ). As discussed in Section 1.6 symmetrizing will give us the finer data which parametrizes individual representations, instead of L-packets. So let

(7.9)  \begin{aligned}P & = P(G, γ) \\ P^{∨} & = P(G^{∨}, γ^{∨}). \end{aligned}

Define P_{c} and P_{c}^{∨} similarly.

Definition 7.10 An L-datum for (G, γ) is a quadruple

(7.11)  S = (ξ, B_{1}, η, B_{1}^{∨})

where S = (ξ, B_{1}) is an L-datum for G, and S^{∨} = (η, B_{1}^{∨}) is an L-datum for G^{∨}. Let θ_{ξ,𝔥} (respectively θ_{η,𝔥^{∨}}) be θ_{ξ} restricted to 𝔥 (resp. θ_{η} restricted to 𝔥^{∨}). Recalling (2.3), we require that these satisfy

(7.12)  (θ_{ξ,𝔥})^{t} = −θ_{η,𝔥^{∨}}.

A complete L-datum is a set

(7.13)  S_{c} = (ξ, B_{1}, η, B_{1}^{∨}, λ)

where the same conditions hold, λ ∈ P_{reg} and exp(2πiλ) = η^{2}.

Let

(7.14) L = {L-data}/G × G^{∨} ⊂ P × P^{∨} L_{c} = {complete L-data}/G × G^{∨} ⊂ P × P_{c}^{∨}.

Suppose S_{c} = (S, S_{c}^{∨}) = (ξ, B_{1}, η, B_{1}^{∨}, λ) is a complete L-datum for (G, γ). By [1, Theorem 2.12] associated to S_{c} is a (𝔤, K_{ξ})-module I(S_{c}). This is a standard module, with regular integral infinitesimal character λ, and has a unique irreducible quotient J(S_{c}). We obtain the following version of the Langlands classification. For more detail, in particular information on how to write J(S_{c}) in various classifications, see [4, Section 9].

Theorem 7.15 ([1], Theorem 2-12) The map

(7.16)  L_{c} ∋ S_{c} ↦ J(S_{c}) ∈ Π(G, γ)

is a bijection.

The large L-packet Π_{S_{c}} of (7.5) is the set of irreducible representations defined by L-data with second coordinate S_{c}^{∨}:

Π_{S_{c}^{∨}} = {J(S) | S = (S, S_{c}^{∨})}.

Recall the right hand side of (7.16) consists of the equivalence classes of pairs (ξ, π) where ξ is a strong involution and π is a (𝔤, K_{ξ})-module. This is a somewhat subtle space. Furthermore, as discussed after Proposition 7.4, we would like to replace L_{c} with L on the left side of 7.16. This amounts to considering λ only up to X^{∗}(H), and using the translation principle. With these considerations in mind we give several alternative formulations of Theorem 7.15.

Suppose G(ℝ) is a real form of G (cf. Remark 3.5) and Λ ⊂ P_{reg}. Recall (cf. 1.1) Π(G(ℝ), λ) is the set of equivalence classes of irreducible admissible representations with infinitesimal character λ. By analogy with Definition 6.3 define Π(G(ℝ), Λ) = ∐_{λ∈Λ} Π(G(ℝ), λ).

Theorem 7.17

(1) Fix a set Λ ⊂ P_{reg} of representatives of P/X^{∗}(H). This is a finite set if G is semisimple. The map (7.16) induces a bijection

(7.18)  L ⟷^{1−1} Π(G, γ, Λ)

(cf. Definition 6.3). If G is semisimple and simply connected we may take Λ = {ρ}, the infinitesimal character of the trivial representation.

(2) Let I ≃ I/G be a set of representatives of the strong real forms as in (5.15). For each i ∈ I let G_{i}(ℝ) be the real form of G corresponding to the strong involution ξ_{i} (cf. Section 3). Choose Λ as in (1). The map (7.16) induces a bijection

(7.19)  L ⟷^{1−1} ∐_{i∈I} Π(G_{i}(ℝ), Λ).

(3) Suppose G is adjoint. Write G_{1}(ℝ), …, G_{m}(ℝ) for the real forms of G in the given inner class, and choose representatives λ_{1}, …, λ_{n} ∈ P_{reg} for P/R. The map (7.16) induces a bijection:

(7.20)  L ⟷^{1−1} ∐_{i,j} Π(G_{i}(ℝ), λ_{j}).

Sections 9 and 10 will be concerned with finding a combinatorial description of the set L. Most of the work involves the one-sided parameter space P(G, γ) (7.9). Before turning to this we give a geometric interpretation of this space.

8 Relation with the flag variety

Recall (Definition 7.6) the one-sided parameter space P = P(G, γ) is the set of conjugacy classes of one-sided L-data, i.e. pairs (ξ, B_{1}) where ξ is a strong involution (Definition 5.5) and B_{1} is a Borel subgroup. The space P has a natural interpretation in terms of the flag variety. We see this by conjugating any pair (ξ, B_{1}) to one with ξ in a fixed set of representatives as in in (5.15). In the next section we will instead conjugate B_{1} to B, and thereby obtain a combinatorial model of P.

Let B be the set of Borel subgroups of G. Then the set of one-sided L-data for (G, γ) is I × B, and

(8.1)(a)  P = (I × B)/G.

Every Borel subgroup is conjugate to B, and B ≃ G/B. For ξ ∈ I let

(8.1)(b)  P[ξ] = {(ξ′, B_{1}) | ξ′ is conjugate to ξ}/G.

Then

(8.1)(c)  P[ξ] ≃ (G/K_{ξ} × G/B)/G

with G acting by left multiplication. It is an elementary exercise to see the map

(8.1)(d)  (ξ′, B_{1}) = (gξg^{−1}, hBh^{−1}) ↦ K_{ξ}(g^{−1}h)B (g, h ∈ G)

gives a bijection

(8.1)(e)  P[ξ] ⟷^{1−1} K_{ξ}\G/B.

As in (5.15) choose a set {ξ_{i} | i ∈ I} of representatives of I/G. Then

(8.1)(f)  P ≃ ∐_{i∈I}(G/K_{i} × G/B)/G

and we see:

Proposition 8.2 There is a natural bijection:

(8.3)  P(G, γ) ⟷^{1−1} ∐_{i∈I} K_{i}\G/B.

9 The One Sided Parameter Space

We now turn to the question of formulating an effective algorithm for computing the space L ⊂ P × P^{∨} (7.14). This mainly comes down to computing the one-sided parameter space P, which we do in this section. We put P and P^{∨} together to define the parameter space Z of representations in Section 10.

We begin by looking for a normal form for one-sided L-data.

Fix basic data (G, γ) as usual and set I = I(G, γ). Recall (cf. Section 8) P = (I ×B)/G. Since every Borel subgroup is conjugate to B, every element of I × B may be conjugated to one of the form (ξ, B). Therefore the map

(9.1)(a)  I ∋ ξ ↦ (ξ, B) ∈ (I × B)/G = P

is surjective. Since B is its own normalizer, we see (ξ, B) is G-conjugate to (ξ′, B) if and only if ξ is B-conjugate to ξ′. So we obtain a bijection

(9.1)(b)  I/B ⟷^{1−1} P

from B-orbits on I to P, sending the B-orbit of ξ ∈ I to the G-orbit of the pair (ξ, B).

Now suppose ξ ∈ I. By ([3, Lemma 6.18], [17]), ξ ∈ Norm_{G^{Γ}}(H_{1}) for some Cartan subgroup H_{1} ⊂ B. There exists b ∈ B such that bH_{1}b^{−1} = H, so bξb^{−1} ∈ N^{Γ} = Norm_{G^{Γ}}(H). If b_{1} is another such element then b_{1} = hb with h ∈ H, and b_{1}ξb_{1}^{−1} = h(bξb^{−1})h^{−1}. Therefore

(9.1)(c)  I/B ≃ (I ∩ N^{Γ})/H.

This gives our primary combinatorial construction:

Definition 9.2 The one-sided parameter space for (G, γ) is the set

(9.3)  \begin{aligned}X (G, γ) & = (I ∩ N^{Γ})/H \\ & = \{ξ ∈ Norm_{G^{Γ}\G}(H) | ξ^{2} ∈ Z(G)\}/H. \end{aligned}

This is the set of strong involutions normalizing H, modulo conjugation by H. If (G, γ) is understood we write X = X (G, γ).

From the preceding discussion we have

(9.4)  X = (I ∩ N^{Γ})/H ≃ I/B ≃ P

and we conclude:

Proposition 9.5 There is a canonical bijection

(9.6)  X ⟷^{1−1} P

taking the H-orbit of an element ξ of I ∩ N^{Γ} to the G-orbit of (ξ, B) in P.

Given x ∈ X, let

(9.7)  X [x] = {x′ | x′ is G-conjugate to x}.

This is a slight abuse of notation: we say x, x′ ∈ X are G-conjugate if ξ, ξ′ are G-conjugate, where ξ, ξ′ are pre-images of x, x′ in I ∩ N^{Γ}, respectively.

By Proposition 9.5 and (8.1)(e) we see

(9.8)  X [x] ≃ K_{ξ}\G/B

where ξ is a preimage of x in I ∩ N^{Γ}.

Choose a set {ξ_{i} | i ∈ I} of representatives of I/G as in (5.15). By Proposition 8.2 and (9.8) we obtain:

Corollary 9.9

(9.10)  X ≃ ∐_{i∈I} K_{i}\G/B.

See Examples 12.20 and 12.25.

We need to understand the structure of X in some detail. We now give more information about it. At the same time we reiterate some earlier definitions and introduce the twisted involutions in the Weyl group.

We fix (G, γ) throughout and drop them from the notation.

Let

(9.11)(a)  N = Norm_{G}(H) ⊂ N^{Γ} = Norm_{G^{Γ}}(H)

and

(9.11)(b)  W = N/H ⊂ W^{Γ} = N^{Γ}/H.

Recall (Definition 5.5)

(9.11)(c)  I = {ξ ∈ G^{Γ}\G | ξ^{2} ∈ Z(G)},

and that G acts on I by conjugation. Let

(9.11)(d)  \begin{aligned}X̃ & = I ∩ N^{Γ} \\ & = \{ξ ∈ N^{Γ}\N | ξ^{2} ∈ Z(G)\}. \end{aligned}

This is the set of strong involutions normalizing H.

Let

(9.11)(e)  X = X̃/H

as in (9.3) (the quotient is by the conjugation action).

The group N acts naturally on X̃ and X; the action of N on X factors to W. This action of W on X corresponds to the cross action of W on characters [26, Definition 8.3.1]. We therefore denote this action ×. That is for w ∈ W and x ∈ X, choose n ∈ N mapping to w, ξ ∈ X̃ mapping to x and define

(9.11)(f)  w × x = image of nξn^{−1} in X .

Every strong involution is conjugate to one in X̃, and we see

(9.11)(g)  X̃/N ≃ X /W.

See Proposition 12.9 for an interpretation of this space.

Let

(9.11)(h)  I_{W} = {τ ∈ W^{Γ}\W | τ^{2} = 1}.

Write p̃: X̃ ↦ W^{Γ} for the restriction of the map N^{Γ} ↦ W^{Γ} to X̃. It is immediate that Im(p̃) ⊂ I_{W}, and p̃: X̃ ↦ I_{W} factors to a map

(9.11)(i)  p : X ↦ I_{W} .

Recall δ is the distinguished element of X (Definition 5.1); use the same notation for its image in I_{W}.

Lemma 9.12 ([21]) The map p : X ↦ I_{W} is surjective.

We prove this later; see Proposition 12.12 and the end of Section 14. For ξ ∈ X̃ the restriction of θ_{ξ} to H only depends on the image x of ξ in

X. Therefore we may define

(9.13)(a)  θ_{x,H} = θ_{ξ} restricted to H.

By Lemma 9.12 I_{W} may be thought of as the set of Cartan involutions of H for this inner class:

(9.13)(b)  I_{W} ⟷^{1−1} {θ_{x,H} | x ∈ X }.

The map ξ ↦ ξ^{2} ∈ Z(G) is constant on fibers of the map X̃ ↦ X. For x ∈ X we define x^{2} ∈ Z(G) accordingly. For z ∈ Z(G) let

(9.13)(c)  X (z) = {x ∈ X | x^{2} = z}.

Note that X (z) is empty unless z ∈ Z^{Γ}.

We can make these constructions more concrete using the distinguished element δ of Definition 5.1. Let θ = int(δ). Then

(9.14)  \begin{aligned}X̃ & = \{x ∈ Nδ | x^{2} ∈ Z(G)\} \\ & = \{gδ | g ∈ N, gθ(g) ∈ Z(G)\} \\ & ⟷^{1−1} \{g ∈ N | gθ(g) ∈ Z(G)\} \\ X & = X̃/\{gδ → hgθ(h^{−1})δ | h ∈ H\} \\ & ⟷^{1−1} \{g ∈ N | gθ(g) ∈ Z(G)\}/\{g → hgθ(h^{−1}) | h ∈ H\} \\ I_{W} & = \{τ ∈ W δ | τ^{2} = 1\} \\ & = \{wδ | w ∈ W, wθ(w) = 1\} \\ & ⟷^{1−1} \{w ∈ W | wθ(w) = 1\} \end{aligned}

The last equality identifies I_{W} with the twisted involutions in the Weyl group. Also note that conjugation in W^{Γ} becomes twisted conjugation:

(9.15)  y · w = ywθ(y^{−1}) (w, y ∈ W).

10 The Space Z and the Main Theorem

We now describe the parameter space for Harish-Chandra modules of strong real forms of G. By Theorem 7.17 we need to describe the set L (Definition 7.10). We have done all of the work describing the one-sided parameter space X (G, γ) (Definition 9.2), and now we merely need to put the two sides together.

Fix basic data (G, γ), and let (G^{∨}, γ^{∨}) be the dual data (cf. Section 4). Let X = X (G, γ) and X^{∨} = X (G^{∨}, γ^{∨}).

Definition 10.1 The two-sided parameter space is the set

(10.2)  Z(G, γ) = {(x, y) ∈ X × X^{∨} | (θ_{x,H})^{t} = −θ_{y,H^{∨}}}.

We may now state the main result on the parametrization of admissible representations of real forms of G. This is an immediate consequence of Theorem 7.17 and the Definitions of L and Z.

Fix a set Λ ⊂ P_{reg} of representatives of P/X^{∗}(H).

Theorem 10.3

(1) There is a natural bijection

(10.4)  Z(G, γ) ⟷^{1−1} Π(G, γ, Λ)

(cf. Definition 6.3).

(2) Let I ≃ I/G be a set of representatives of the strong real forms as in (5.15). For each i ∈ I let G_{i}(ℝ) be the real form of G corresponding to the strong involution ξ_{i} (cf. Section 3). There is a natural bijection

(10.5)  Z(G, γ) ⟷^{1−1} ∐_{i∈I} Π(G_{i}(ℝ), Λ).

(3) Suppose G is adjoint. Write G_{1}(ℝ), …, G_{m}(ℝ) for the (equivalence classes of) real forms of G in the given inner class, and choose representatives λ_{1}, …, λ_{n} ∈ P_{reg} for P/R. There is a natural bijection:

(10.6)  Z(G, γ) ⟷^{1−1} ∐_{i,j} Π(G_{i}(ℝ), λ_{j}).

For some examples see the end of Section 12. Many more examples are worked out in detail in [2].

We note that Z(G, γ) can be viewed as a space of orbits as follows. Let I (resp. I^{∨}) be a set of representatives of I/G (resp. I^{∨}/G^{∨}), as in (5.15)(a). For i ∈ I (resp. j ∈ I^{∨}) let K_{i} (resp. K_{j}^{∨}) be as in (5.15)(b). Then

(10.7)  Z ⊂ ∐_{i∈I} K_{i}\G/B × ∐_{j∈I^{∨}} K_{j}^{∨}\G^{∨}/B^{∨}.

We now give the statement of Vogan Duality [27] in this setting. It is evident that the definition of Z(G, γ) is entirely symmetric in G and G^{∨}: the map (x, y) ↦ (y, x) is a bijection between Z(G, γ) and Z(G^{∨}, γ^{∨}).

Corollary 10.8 (Vogan Duality) Fix a set of representatives Λ ⊂ P_{reg} of P (G, H)/X^{∗}(H), and a set Λ^{∨} ⊂ P_{reg}^{∨} of representatives of P (G^{∨}, H^{∨})/X^{∗}(H^{∨}). There is a natural bijection

(10.9)  Π(G, γ, Λ) ↔ Π(G^{∨}, γ^{∨}, Λ^{∨}).

This bijection is compatible, in a precise sense, with the duality of [27, Theorem 13.13]. See [1] and [3] for details. See the Table in example 12.20 for the case of SL(2)/P SL(2), and [2] for some more elaborate examples.

11 Fibers of the map p : X → I

In Sections 11 through 14 we study the space X in more detail, and relate it to structure and representation theory of real groups. We begin with a study of the fibers of p̃ and p. We work in the setting of Section 9.

Fix τ ∈ I_{W}. Let X̃_{τ} = p̃^{−1}(τ) and X_{τ} = p^{−1}(τ). For z ∈ Z(G) let X_{τ}(z) = X_{τ} ∩ X (z) = {x ∈ X | p(x) = τ, x^{2} = z}. Let

(11.1)  \begin{matrix}H′_{−τ} = & \{h ∈ H | hτ(h) ∈ Z(G)\} \\ H_{−τ} = & \{h ∈ H | hτ(h) = 1\} \\ A_{τ} = & \{hτ(h^{−1}) | h ∈ H\}. \end{matrix}

Note that A_{τ} is a connected torus, and is the identity component of H_{−τ}.

Proposition 11.2

(1) H′_{−τ} acts simply transitively on X̃_{τ},

(2) H′_{−τ}/A_{τ} acts simply transitively on X_{τ},

(3) Fix z ∈ Z(G). If X_{τ}(z) is non-empty then H_{−τ}/A_{τ} acts simply transitively on X_{τ}(z). If z ∉ Z(G)^{Γ} (the Γ-invariants of Z(G)) then X_{τ}(z) is empty.

In particular |X_{τ}(z)| is a power of 2 (or 0). If Z(G)^{Γ} is finite then X is a finite set.

Proof. Choose ξ ∈ X̃_{τ}. Then X̃_{τ} = {hξ | h ∈ H, (hξ)^{2} ∈ Z(G)} = {hξ | hτ(h)ξ^{2} ∈ Z(G)}. The first claim follows.

For h ∈ H we have hξh^{−1} = hτ(h^{−1})ξ. This shows that the stabilizer in H′_{−τ}, for the left multiplication action of H, of the image of ξ in X is {hτ(h^{−1}) | h ∈ H} = A_{τ}. This proves (2), and (3) follows immediately from the fact that (hξ)^{2} = hτ(h)ξ^{2}, and the fact that ξ^{2} ∈ Z(G)^{Γ}. The assertion about |X_{τ}(z)| is clear, since H_{−τ}/A_{τ} is an elementary abelian two-group.

By (3) X is the union of the finite sets X_{τ}(z) for τ ∈ I_{W} and z ∈ Z(G)^{Γ}, and the final conclusion follows. □

Remark 11.3 Let T_{τ} be the identity component of the fixed points of τ acting on H. Then T_{τ} and A_{τ} are connected tori, H = T_{τ}A_{τ} and A_{τ} ∩ T_{τ} is an elementary abelian two group. The group in Proposition 11.2 (3) is

(11.4)  H_{−τ}/A_{τ} ≃ T_{τ}(2)/A_{τ} ∩ T_{τ}.

If we write the real torus corresponding to τ as (ℝ^{×})^{a} × (S^{1})^{b} × (ℂ^{×})^{c} then T_{τ}(2) ≃ (ℤ/2ℤ)^{b+c}, A_{τ} ∩ T_{τ} ≃ (ℤ/2ℤ)^{c} and T_{τ}(2)/A_{τ} ∩ T_{τ} ≃ (ℤ/2ℤ)^{b}.

Remark 11.5 It is helpful to note that

(11.6)  H_{−τ}/A_{τ} ≃ [H^{∨}(ℝ)/H^{∨}(ℝ)^{0}]^{∧}

where H^{∨} is the dual torus to H, with real form H^{∨}(ℝ) defined by the Cartan involution −τ^{∨}.

12 Action of W on X

We now study the action of W on X, which plays an important role. We begin with some definitions and terminology.

Fix τ ∈ I_{W}. Let

(12.1)

Δ_{i} = {α ∈ Δ | τ(α) = α} (the imaginary roots)
Δ_{r} = {α ∈ Δ | τ(α) = −α} (the real roots)
Δ_{cx} = {α ∈ Δ | τ(α) ≠ ±α} (the complex roots)
Δ_{i}^{+} = Δ_{i} ∩ Δ^{+}, Δ_{r}^{+} = Δ_{r} ∩ Δ^{+}
W_{i} = W (Δ_{i})
W_{r} = W (Δ_{r}).

We will write Δ_{i,τ}, W_{i,τ} etc. to indicate the dependence on τ. We also refer to the τ-imaginary, τ-real roots, etc.

Let ρ_{i} = \frac{1}{2} ∑_{α∈Δ_{i}^{+}} α, and ρ_{r}^{∨} = \frac{1}{2} ∑_{α∈Δ_{r}^{+}} α^{∨}. As in [27, Proposition 3.12] let

(12.2)  \begin{aligned}Δ_{C} & = \{α ∈ Δ | ⟨ρ_{i}, α^{∨}⟩ = ⟨α, ρ_{r}^{∨}⟩ = 0\} ⊂ Δ_{cx} \\ W_{C} & = W (Δ_{C}) \end{aligned}

This is a complex root system.

Now τ acts on W, and we let W^{τ} be the fixed points. By [27, Proposition 3.12]

(12.3)  W^{τ} = (W_{C})^{τ} ⋉ (W_{i} × W_{r}).

Note that W_{i} and W_{r} are Weyl groups of the root systems Δ_{i} and Δ_{r} respectively; also (W_{C})^{τ} is isomorphic to the Weyl group of the root system (Δ_{C})^{τ} [27].

Fix ξ ∈ X̃_{τ} and let θ = θ_{ξ}, K = K_{ξ}. For α ∈ Δ_{i} let X_{α} be an αroot vector. Then θ_{ξ}(X_{α}) only depends on the image x of ξ in X. We say gr_{x}(α) = 0 if θ_{ξ}(X_{α}) = X_{α} and 1 if θ_{x}(X_{α}) = −X_{α}. This is a ℤ/2ℤ-grading of Δ_{i} in the sense that if α, β, α + β ∈ Δ_{i} then gr_{x}(α + β) = gr_{x}(α) + gr_{x}(β) (mod 2). These are the compact and noncompact imaginary roots.

Let W (K, H) = Norm_{K}(H)/H∩K. This is isomorphic to W (G(ℝ), H(ℝ)) where G(ℝ) is the real form of G corresponding to θ, and we call it the real Weyl group. (This contains the Weyl group of the root system of real roots.) Clearly W (K, H) ⊂ W^{τ}. Let M = Cent_{G}(A_{τ}) (cf. 11.1). By [27, Proposition 4.16],

(12.4)  W (K, H) = (W_{C})^{τ} ⋉ (W (M ∩ K, H) × W_{r}).

We have

(12.5)  W_{i,c} ⊂ W (M ∩ K, H) ≃ W_{i,c} ⋉ A(H) ⊂ W_{i}

where W_{i,c} is the Weyl group of the compact imaginary roots (i.e. gr_{ξ}(α) = 0) and A(H) is a certain two-group [27]. This describes W (K, H) in terms of the Weyl groups (W_{C})^{τ}, W_{r} and W_{i,c}, which are straightforward to compute, and the two-group A(H). For more information on W (K, H) see Proposition 12.14.

Let

(12.6)  H = {(ξ, H_{1}) | ξ ∈ I, H_{1} a θ_{ξ}-stable Cartan subgroup}/G.

With I, θ_{i} and K_{i} as in (5.15) we may conjugate ξ to some ξ_{i}, and this shows

(12.7)  H ≃ ∐_{i∈I} {θ_{i}-stable Cartan subgroups of G}/K_{i}.

On the other hand every Cartan subgroup is conjugate to H, and the normalizer of H is N, so

(12.8)  H ≃ I ∩ N^{Γ}/N = X̃/N ≃ X /W

(cf. (9.11)(g)).

With notation as in Corollary 9.9 for i ∈ I we may assume ξ_{i} ∈ X̃, and define x_{i} ∈ X and X_{i} = X [x_{i}] accordingly. We conclude

Proposition 12.9 For each i ∈ I we have

(12.10)  X_{i}/W ↔ {θ_{i}-stable Cartan subgroups of G}/K_{i}.

Taking the union over i ∈ I gives

(12.11)  X /W ↔ ∐_{i} {θ_{i}-stable Cartan subgroups of G}/K_{i}.

Recall that by (9.8) and Proposition 8.2 X_{i} ≃ K_{i}\G/B.

Proposition 12.12 ([21]) The map p : X_{i}/W ↦ I_{W} /W is injective. If θ_{i} is quasisplit it is a bijection.

Remark 12.13 This says that the conjugacy classes of Cartan subgroups of any real form of G embed in those of the quasisplit form. See [20, page 340].

Proof. For injectivity we have to show that ξ, ξ′ ∈ X̃, p̃(ξ) = p̃(ξ′) and ξ′ = gξg^{−1} (g ∈ G) implies ξ′ = nξn^{−1} for some n ∈ N. The condition p̃(ξ) = p̃(ξ′) implies ξ′ = hξ for some h ∈ H, so gξg^{−1} = hξ, i.e. gθ_{ξ}(g^{−1}) = h. By [21, Proposition 2.3] there exists n ∈ N satisfying h = nθ_{ξ}(n^{−1}), and then ξ′ = nξn^{−1}.

We defer the proof of surjectivity in the quasisplit case to the end of Section 14. □

The real Weyl group (see the discussion preceding (12.4)) appears naturally in our setting. Fix ξ ∈ X̃, let K = K_{ξ}, and let x be the image of ξ in X.

Proposition 12.14 W (K, H) ≃ Stab_{W} (x).

Proof. We have

(12.15)

W (K, H) = Norm_{K}(H)/H ∩ K
= Stab_{N}(ξ)/Stab_{H}(ξ)
= Stab_{N}(ξ)H/H.

It is easy to see that Stab_{N}(ξ)H = Stab_{N}(x), so this equals

Stab_{N}(x)/H ≃ Stab_{N/H}(x) = Stab_{W} (x).
□

Now fix τ ∈ I_{W}. By Proposition 12.14 and (12.4) we see (W_{C})^{τ} and W_{r} act trivially on X_{τ}. It is worth noting that we can see this directly.

Proposition 12.16 Both (W_{C})^{τ} and W_{r} act trivially on X_{τ}.

This proof was communicated to us by David Vogan. Proof. Fix ξ ∈ X̃_{τ}. The group (W_{C})^{τ} is generated by elements s_{α}s_{τα} where α ∈ Φ_{C}. So suppose α ∈ Φ_{C} and let σ_{α} ∈ N be a preimage of s_{α} ∈ W. Let σ_{τ(α)} = ξσ_{α}ξ^{−1}. Note that α + τ(α) is not a root, since it would have to be imaginary, and (by (12.2)) orthogonal to ρ_{i}. Therefore the root subgroups G_{α} and G_{τ(α)} commute. Then ξσ_{α}σ_{τ(α)}ξ^{−1} = σ_{τ(α)}σ_{α} = σ_{α}σ_{τ(α)}.

If α is a τ-real root this reduces easily to a computation in SL(2). We omit the details. □

Fix τ ∈ I_{W} and suppose x, x′ ∈ X_{τ}. As a consequence of Propositions 12.12 and 12.14 we have

(12.17)  x′ is G-conjugate to x ⇔ x′ = w × x for some w ∈ W_{i,τ}.

Another useful result obtained from the action of W^{τ} is the computation of strong real forms.

Proposition 12.18 Every element x ∈ X is G-conjugate to an element of X_{δ}, and there is a canonical bijection between X_{δ}/W_{i,δ} and the set of strong real forms of (G, γ).

(See the remark after (9.7) for the notion of G-conjugacy.) This corresponds to the fact that every real form in the given inner class contains a fundamental (i.e. most compact) Cartan subgroup. Note that if G is adjoint this gives a bijection between X_{δ}/W_{i,δ} and real forms in the given inner class.

We defer the proof until the end of Section 14. See Examples 12.20 and 12.25.

12.1 Recapitulation

We summarize the main results on the translation between the structure of X and some standard objects for G.

Fix basic data (G, γ) as in Section 4 and define G^{Γ} as in Definition 5.1. Let X = X (G, γ) (cf. (9.11)(e)) and define I_{W} as in (9.11)(h). Fix a set {ξ_{i} | i ∈ I} of representatives of the strong real forms, i.e. I/G, as in (5.15), and for i ∈ I let θ_{i} = θ_{ξ_{i}} and K_{i} = K_{ξ_{i}}. Recall δ is the distinguished element of G^{Γ}, or its image in I_{W}. Also W_{i,δ} is the Weyl group of the δ-imaginary roots. Let θ_{qs} be a quasisplit involution in this inner class (cf. Lemma 5.16) and let K_{qs} = G^{θ_{qs}}.

Proposition 12.19 We have bijections:

(1) X ⟷^{1−1} ∐_{i∈I} K_{i}\G/B (Corollary 9.9),

(2) X_{δ}/W_{i,δ} ⟷^{1−1} {strong involutions}/G={strong real forms} (Prop. 12.18),

(3) I_{W} /W ⟷^{1−1} {θ_{qs}-stable Cartan subgroups in G}/K_{qs} (Prop. 12.12),

(4) X /W ⟷^{1−1} ∐_{i}{θ_{i}-stable Cartan subgroups in G}/K_{i} (Prop. 12.9),

(5) X_{τ}(z) ≃ [H^{∨}(ℝ)/H^{∨}(ℝ)^{0}]^{∧} or is empty (τ ∈ I_{W} , z ∈ Z,H^{∨}(ℝ) as in Remark 11.5),

(6) Stab_{W} (x) ≃ W (K_{ξ}, H) (ξ ∈ X̃, with image x ∈ X) (Prop. 12.14).

Example 12.20 We illustrate each part of the Proposition in the case of SL(2, ℂ). In this case Out(G) = 1 so there is only one inner class of involutions, and we drop δ from the notation. See examples 3.4 and 5.6.

Write H = \{diag(z, \frac{1}{z}) | z ∈ ℂ^{×}\}, W = \{1, s\}, and let t = diag(i, −i). Let n be any element of Norm_{G}(H) mapping to s ∈ W. Then Norm_{G}(H) = H ∪ Hn, and

(12.21)  \begin{aligned}X̃ & = \{ξ ∈ H ∪ Hn | ξ^{2} = ±Id\} \\ & = \{±Id, ±t\} ∪ Hn \\ X & = \{±Id, ±t\}/H ∪ Hn/H \end{aligned}

Note that H acts trivially on {±Id, ±t}, and hnh^{−1} = h^{2}n for all h ∈ H, so Hn/H is a singleton. Therefore

(12.22)  X = {±Id, ±t, n}.

Strictly speaking these are elements of X̃ representing X.

Since δ = 1 we have X_{δ} = {±Id, ±t}, and W_{i,δ} = W. Part (2) of the Proposition says we can take I to be a set of representatives of X_{δ}/W = {±Id, t}. Recall (5.6) we think of these as “strong real forms” SU(2, 0), SU(0, 2) and SU(1, 1) ≃ SL(2, ℝ), respectively. Then

(12.23)  X [Id] = {Id}, X [−Id] = {−Id}, X [t] = {t, −t, n}.

Now G/B is isomorphic to the complex projective plane ℂ ∪ {∞}. We have K_{±Id} = G and K_{±Id}\G/B is a point. We label these orbits O_{2,0} and O_{0,2}, respectively. On the other hand K_{t} ≃ ℂ^{×}, which acts on G/B by z : u ↦ z^{2}u. Therefore there are three orbits of this action: O_{0} = {0}, O_{∞} = {∞} and O_{∗} = ℂ^{×}.

So the bijection of (1) is

(12.24)

x ∈ X Id -Id t -t n
K_{x}  G  G ℂ^{×} ℂ^{×} ℂ^{×}
Orbit O_{2,0} O_{0,2} O_{0} O_{∞} O_{∗}

In this case I_{W} = W = {1, s}. The quasisplit group is SL(2, ℝ), which has two conjugacy classes of Cartan subgroups. The compact Cartan subgroup T ≃ S^{1} corresponds to 1 ∈ W, and the split Cartan subgroup A ≃ ℝ^{×} corresponds to s ∈ W. This is (3) in this case.

Now X /W has four elements I, −I, t and n, corresponding to the compact Cartan subgroups of SU(2, 0), SU(0, 2), SU(1, 1), and the split Cartan subgroup of SU(1, 1), respectively. This is the content of (4).

For (5) we have X_{1}(Id) = {±Id} and X_{1}(−Id) = {±t}. In this case H(ℝ) ≃ S^{1}, and H^{∨}(ℝ) ≃ ℝ^{×}, so H^{∨}(ℝ)/H^{∨}(ℝ)^{0} ≃ ℤ/2ℤ. On the other hand X_{s}(Id) = ∅ and X_{s}(−Id) = {n}. In this case H(ℝ) = ℝ^{×} and H^{∨}(ℝ) = S^{1} is connected.

Finally consider (6). We have Stab_{W} (±Id) = W, and Stab_{W} (±t) = 1. This corresponds to the fact that W (SU(2), S^{1})) = W, and W (SL(2, ℝ), S^{1}) = 1. On the other hand Stab_{W} (n) = W, i.e. W (SL(2, ℝ), ℝ^{×}) = W.

Most of the conclusions of the Proposition seen in Figure 1. Projection p : X ↦ I_{W} is written vertically.

SU(2, 0)  Id  } ; | ⟶ z = Id SU(0, 2)  −Id { ; SU(1, 1)  t  } ; | ⟶ z = −Id −t  w I_{W}  1  s_{α} Cartan T  A

Figure 1: X for SL(2)

Example 12.25 We reconsider the previous example with P SL(2, ℂ) in place of SL(2, ℂ). Again γ = 1 and we drop it from the notation.

Recall P SL(2, ℂ) ≃ SO(3, ℂ), and it is easier to work with the latter real-

ization, with respect to the form (\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}). We take H = \{diag(z, \frac{1}{z}, 1) | z ∈

ℂ^{×}}, write W = {1, s} and let n be a representative in Norm_{G}(H) of s. Let t = diag(−1, −1, 1).

As in the previous example we have:

(12.26)  \begin{aligned}X & = \{ξ ∈ H ∪ Hn | ξ^{2} = Id\}/H \\ & = \{Id, t\}/H ∪ (Hn)/H \\ & = \{Id, t, n\}. \end{aligned}

In this case we can take I = {Id, t}, and

(12.27)  X [Id] = {Id}, X [t] = {t, n}.

Then K_{Id} = G and K_{Id}\G/B is a point. On the other hand K_{t} ≃ O(2, ℂ), which has two orbits on the projective plane: {0, ∞} and ℂ^{×}. This is (1) of the Proposition in this case.

In this case (2) says that X_{1}/W = {Id, t}, corresponding to the two real forms of G.

The analogue of (12.24) is

x ∈ X Id t n
K_{x}  G ℂ^{×} ℂ^{×}
Orbit O_{2,0} O_{0} O_{∗}

Statement (3) is the same as for SL(2, ℂ): I_{W} /W has two elements, corresponding to the two conjugacy classes of Cartan subgroups of SO(2, 1).

For (4), X /W = X = {Id, t, n}; corresponding to the compact Cartan subgroups of SO(3), SO(2, 1), and the split Cartan subgroup of SO(2, 1), respectively.

Next, X_{1}(Id) = {Id, t} and X_{s}(I) = {n}, corresponding to H^{∨}(ℝ) = ℝ^{×} and S^{1} as in the previous example. This gives (5).

Finally note that Stab_{W} (1) = Stab(n) = W as in the previous example. However Stab_{W} (t) = W; i.e. W (SO(2, 1), S^{1}) = W. Comparing this case with the fact that Stab_{W} (t) = 1 in the case of SL(2, ℂ) illustrates how Stab_{W} (ξ) depends in a subtle way on isogenies. See (12.5).

Example 12.28 Having described X for SL(2, ℂ) and P SL(2, ℂ) we can now describe the representation theory of real forms of these groups in terms of the space Z. See Theorem 10.3, and Examples 12.20 and 12.25. Note that the representations are parametrized by pairs of orbits as in (10.7).

Write ℂ for the trivial representation.

For SU(1, 1) ≃ SL(2, ℝ), at infinitesimal character ρ, write DS_{±} for the discrete series representations and P S_{odd} for the irreducible (non-spherical) principal series representation.

Consider SO(2, 1). Let sgn be the sign representation of SO(2, 1), and DS be the unique discrete series representation with infinitesimal character ρ. At infinitesimal character 2ρ SO(2, 1) has two irreducible principal series representations denoted P S_{±}.

Table of representations of SL(2) and P GL(2)

Orbit x x^{2} θ_{x} G_{x}(ℝ) λ rep  Orbit y y^{2} θ_{y} G_{y}^{∨}(ℝ) λ rep
O_{2,0}  Id Id 1 SU(2, 0) ρ ℂ  O′_{∗}  n Id -1 SO(2, 1) 2ρ P S_{+}
O_{0,2} -Id Id 1 SU(0, 2) ρ ℂ  O′_{∗}  n Id -1 SO(2, 1) 2ρ P S_{−}
O_{0}  t -Id 1 SU(1, 1) ρ DS_{+} O′_{∗}  n Id -1 SO(2, 1) ρ ℂ
O_{∞} -t -Id 1 SU(1, 1) ρ DS_{−} O′_{∗}  n Id -1 SO(2, 1) ρ sgn
O_{∗}  n -Id 1 SU(1, 1) ρ ℂ  O′_{+}  t Id -1 SO(2, 1) ρ DS
O_{∗}  n Id 1 SU(1, 1) ρ P S_{odd} O′_{3,0}  Id Id 1 SO(3) ρ ℂ

See [2] for more detail on this example.

13 The reduced parameter space

If G is adjoint we saw in Section 10 that the parameter space X = X (G, γ) is perfectly suited to parametrizing representations of real forms of G. If G is not adjoint then strong involutions play an essential role, and the difference between involutions and strong involutions is unavoidable. Nevertheless in some respects the space X is larger than necessary, and a satisfactory theory is obtained with a smaller set, the reduced one-sided parameter space. While X may be infinite, this is always a finite set.

The most economical possibility would be to keep a single orbit of strong involutions over each orbit of real forms. However there is no canonical way to make this choice. The reduced parameter space provides a more canonical way to reduce to a small finite number of choices related to the center.

Let Z = Z(G), and recall (cf. 11.2(3)) Z^{Γ} is the Γ-invariants in Z. There is a natural action of Z on X by left multiplication. This preserves the fibers X_{τ}, and commutes with the conjugation action of G. For x ∈ X recall (9.7) X [x] is the set of elements of X conjugate to x. If z ∈ Z multiplication by z is a bijection:

(13.1)  X [x] ⟷^{1−1} X [zx]

(cf. (9.7)). In other words the orbit pictures for x and zx are identical. Suppose ξ ∈ X̃ lies over x. Harish-Chandra modules for ξ are (𝔤, K_{ξ}) modules; since K_{ξ} = K_{zξ}, Harish-Chandra modules for ξ are exactly the same as Harish-Chandra modules for zξ, with the same notion of equivalence.

For example suppose G = SL(2) and take ξ = I and z = −I. Then ξ = I and zξ = −I both correspond to the compact group SU(2). See Example 5.6.

Write θ for the action of the non-trivial element of Γ on Z. For z ∈ Z recall X (z) is the set of elements of X satisfying x^{2} = z (9.13)(c), and X (z) is empty unless z ∈ Z^{Γ}. If x ∈ X (z′) and z ∈ Z then

(13.2)  zx ∈ X (z′zθ(z)).

It is easy to see that

(13.3)  Z^{Γ}/{zθ(z) | z ∈ Z} ≃ H^{2}(Γ, Z)

is a finite set. This comes down to the fact that if Z is a torus then Z^{Γ}/{zθ(z)} ≃ (ℤ/2ℤ)^{n} where n is the number of ℝ^{×} factors in the corresponding real torus (cf. Remark 11.3).

Definition 13.4 Choose a set of representatives Z_{0} ⊂ Z^{Γ} for Z^{Γ}/{zθ(z)} ≃ H^{2}(Γ, Z). The reduced parameter space is

(13.5)  X^{r}(G, γ) = ∐_{z∈Z_{0}} X (z).

Example 13.6 Let G = SL(n, ℂ), and let γ = 1. Suppose p + q = n and α^{n} = (−1)^{q}. Let

ξ_{α} = diag(\overbrace{α, …, α}^{p}, \overbrace{−α, …, −α}^{q})

These are representatives of the equivalence classes of strong involutions in this inner class. For fixed p ≠ q there are n strong real forms, all mapping to the real form SU(p, q). In other words we obtain n identical orbit pictures. If p = q a similar statement holds, except that ξ_{α} is conjugate to ξ_{−α}.

If n is odd then Z^{Γ}/{zθ(z)} = Z/Z^{2} is trivial, so we take Z_{0} = {I}, and the equivalence classes of strong involutions in X_{0}(G, γ) are represented by

(13.7)  (\overbrace{1, …, 1}^{p}, \overbrace{−1, …, −1}^{q}) (q even).

For each p, q there is a unique strong involution mapping to the real form SU(p, q), instead of n as we had earlier.

If n is even then Z^{Γ}/{zθ(z)} has order 2. We can take Z_{0} = {I, ζI} where ζ is a primitive n^{th} root of 1. Let τ be a primitive 2n^{th} root of 1. Then the strong real forms in X_{0}(G, γ) are

(13.8)

±diag(\overbrace{1, …, 1}^{p}, \overbrace{−1, …, −1}^{q}) (q even, p ≠ q)

diag(\overbrace{1, …, 1}^{p}, \overbrace{−1, …, −1}^{p})

±diag(\overbrace{τ, …, τ}^{p}, \overbrace{−τ, …, −τ}^{q}) (q odd).

In this case there are two strong real forms mapping to the real form SU(p, q) if p ≠ q, and 1 if p = q.

The calculations needed to understand representation theory (see Section 10) take place entirely in a fixed set X (z). The sets X (z′) and X (z′zθ(z)) are canonically identified, so it is safe to think of X (z) as being defined for z ∈ Z_{0}. The atlas software takes this approach.

14 Cayley Transforms and the Cross Action

We continue to work with the one-sided parameter space X = X (G, γ). We begin with some formal constructions.

Fix x ∈ X and let τ = p(x) ∈ I_{W}. Recall (Section 12) τ defines the real, imaginary and complex roots, and x defines a grading gr_{x} of the imaginary roots. Suppose α is an imaginary noncompact root, i.e. τ(α) = α and gr_{x}(α) = 1.

Let G_{α} be the derived group of Cent_{G}(ker(α)), and H_{α} ⊂ G_{α} the oneparameter subgroup corresponding to α. Then G_{α} is isomorphic to SL(2) or P SL(2) and H_{α} is a Cartan subgroup of G_{α}. Choose σ_{α} ∈ Norm_{G_{α}}(H_{α})\H_{α}; then σ_{α}(α) = −α. Let m_{α} = σ_{α}^{2} = α^{∨}(−1).

Definition 14.1 Suppose x ∈ X and α is a noncompact imaginary root. Choose a pre-image ξ of x in X̃, and define c^{α}(x) to be the image of σ_{α}ξ in X.

Lemma 14.2

(1) c^{α}(x) is well defined, independent of the choices of σ_{α} and ξ.

(2) c^{α}(x) is G-conjugate to x, and c^{α}(x)^{2} = x^{2}.

(3) p(c^{α}(x)) = s_{α}p(x) ∈ I_{W}.

(Here and in Lemma 14.9 keep in mind the remark after (9.7) regarding the notion of G-conjugacy.) Proof. Choose ξ, and let t = α^{∨}(i) ∈ H_{α}. Suppose h ∈ H_{α}. We have a few elementary identities, essentially in SL(2):

(14.3)  \begin{aligned}σ_{α}hσ_{α}^{−1} & = h^{−1}, hσ_{α}h^{−1} = h^{2}σ_{α} \\ ξhξ^{−1} & = h \\ tgt^{−1} & = ξgξ^{−1} (g ∈ G_{α}) \\ ξσ_{α}ξ^{−1} & = σ_{α}^{−1}. \end{aligned}

The first two lines follow from σ_{α}(α^{∨}) = −α^{∨} and θ_{ξ}(α^{∨}) = α^{∨}. For the third, int(t) and θ_{ξ} agree on G_{α}, since they agree on H_{α} and the ±α root spaces. The final assertion follows from the third and a calculation in SL(2).

Now σ_{α}ξ clearly normalizes H, and

(σ_{α}ξ)^{2} = σ_{α}(ξσ_{α}ξ^{−1})ξ^{2} = ξ^{2} ∈ Z(G),

so σ_{α}ξ ∈ X̃.

Given a choice of σ_{α} any other choice is of the form h^{2}σ_{α} for some h ∈ H_{α}, and

(14.4)  (h^{2}σ_{α})ξ = (hσ_{α}h^{−1})ξ = hσ_{α}(h^{−1}ξh)h^{−1} = h(σ_{α}ξ)h^{−1}.

Therefore the image of σ_{α}ξ in X is independent of the choice of σ_{α}.

We need to show that σ_{α}ξ and σ_{α}hξh^{−1} have the same image in X for all h ∈ H. Write H = H_{α}(ker(α)). If h ∈ H_{α} then hξh^{−1} = ξ so the assertion is obvious. If h ∈ ker(α) then σ_{α}h = hσ_{α}, and σ_{α}(hξh^{−1}) = h(σ_{α}ξ)h^{−1}.

For the second assertion, we actually show c^{α}(x) is conjugate to x by an element of G_{α}. By a calculation in SL(2) it is easy to see g(σ_{α}t)g^{−1} = t for some g ∈ G_{α}. Therefore

(14.5) g(σ_{α}ξ)g^{−1} = g(σ_{α}tt^{−1}ξ)g^{−1} = g(σ_{α}t)g^{−1}g(t^{−1}ξ)g^{−1} = tt^{−1}ξ = ξ.

The fact that c^{α}(x)^{2} = x^{2} follows immediately, and the final assertion is obvious. □

We now define inverse Cayley transforms. Suppose ξ ∈ X̃, and let τ = p̃(ξ). Suppose α is a τ-real root. Define G_{α} and H_{α} as before. Let m_{α} = α^{∨}(−1).

Lemma 14.6 There exists σ_{α} ∈ Norm_{G_{α}}(H_{α})\H_{α} so that σ_{α}ξ = gξg^{−1} for some g ∈ G_{α}. The only other element satisfying these conditions is m_{α}σ_{α}.

Proof. This is similar to the previous case. The involution θ_{ξ} restricted to G_{α} is inner for G_{α}, and acts by h ↦ h^{−1} for h ∈ H_{α}. Therefore we may choose y ∈ Norm_{G_{α}}(H_{α})\H_{α} so that ygy^{−1} = ξgξ^{−1} for all g ∈ G_{α}. By a calculation in SL(2) we may choose σ_{α} so that g(σ_{α}y)g^{−1} = y for some g ∈ G_{α}. Then

(14.7) g(σ_{α}ξ)g^{−1} = g(σ_{α}yy^{−1}ξ)g^{−1} = g(σ_{α}y)g^{−1}g(y^{−1}ξ)g^{−1} = yy^{−1}ξ = ξ.

We have σ_{α}y ∈ H_{α}, and α(σ_{α}y) = −1. Therefore any two such choices differ by m_{α}. □

Definition 14.8 Suppose ξ ∈ X̃ and α is a real root with respect to θ_{ξ}. Let c_{α}(ξ) = {σ_{α}ξ, m_{α}σ_{α}ξ}.

If x ∈ X choose ξ ∈ X̃ mapping to x, and define c_{α}(x) to be the image of c_{α}(ξ) in X. This is a set with one or two elements.

The analogue of Lemma 14.2 is immediate:

Lemma 14.9 Suppose x ∈ X and α is a real root with respect to θ_{x}.

(1) c_{α}(x) is well defined, independent of the choice of ξ.

(2) If y ∈ c_{α}(x) then y is G-conjugate to x, and y^{2} = x^{2}.

(3) p(c_{α}(x)) = s_{α}p(x) ∈ I_{W}.

We deduce some simple formal properties of Cayley transforms. Fix τ ∈ I_{W}. If α is τ-imaginary let

(14.10)

X_{τ}(α) = {x ∈ X_{τ} | α is noncompact with respect to θ_{x}}
= {x ∈ X_{τ} | gr_{x}(α) = 1}
= {x ∈ X_{τ} | c^{α}(x) is defined}.

Lemma 14.11

(1) If τ(α) = −α then for all x ∈ X_{τ}, c^{α}(c_{α}(x)) = x,

(2) If τ(α) = α and x ∈ X_{τ}(α) then c_{α}(c^{α}(x)) = {x, m_{α}x}.

(3) The map c^{α} : X_{τ}(α) ↦ X_{s_{α}τ} is surjective, and at most two-to-one

(4) Suppose α is imaginary. The following conditions are equivalent:

(a) c^{α} : X_{τ}(α) → X_{s_{α}τ} is a bijection;

(b) c_{α} : X_{s_{α}τ} → X_{τ}(α) is a bijection;

(c) c_{α}(x) is single valued for all x ∈ X_{s_{α}τ};

(d) m_{α} ∈ A_{τ} (cf. (11.1));

(e) s_{α} ∈ W (K_{ξ}, H) for all ξ ∈ X̃ with image in X_{τ}(α),

(f) x = m_{α}x for all x ∈ X_{τ}(α).

If these conditions fail then c^{α} : X_{τ}(α) → X_{s_{α}τ} is two to one, and c_{α}(x) is double valued for all x ∈ X_{s_{α}τ}.

(5) Suppose α is imaginary with respect to τ. If there exists h ∈ H′_{−τ} (cf. (11.1)) such that α(h) = −1 then X_{τ} is the disjoint union of X_{τ}(α) and hX_{τ}(α). Otherwise X_{τ}(α) = X_{τ}.

We leave the straightforward proof to the reader.

Remark 14.12 Using this Lemma it is straightforward to compute the space X, starting with X_{δ}, and computing the fibers X_{τ} inductively. This shows that it is in fact easier to understand the entire space X rather than the individual pieces K_{i}\G/B ⊂ X (cf. Corollary 9.9). We describe the latter in more detail in the next section.

It is important to understand the effect of Cayley transforms on the grading of the imaginary roots. This is due to Schmid [22]; also see [27, Definition 5.2 and Lemma 10.9].

Lemma 14.13 Suppose τ ∈ I_{W} and x ∈ X_{τ}(α). Then the s_{α}τ-imaginary roots are the τ-imaginary roots orthogonal to α, and for such a root β,

(14.14)  gr_{c^{α}x}(β) = \begin{cases}gr_{x}(β) & if α + β is not a root \\ gr_{x}(β) + 1 & if α + β is a root. \end{cases}

Remark 14.15 Choose a pre-image ξ of x in X̃. By (14.3) we have

(14.16)  ξσ_{α}ξ^{−1} = \begin{cases}σ_{α} & α compact \\ m_{α}σ_{α} & α noncompact. \end{cases}

If (the derived group of) G is simply connected then m_{α} ≠ 1, so gr_{x}(α) = 0 if ξσ_{α}ξ^{−1} = σ_{α}, and 1 otherwise. A calculation in rank 2 shows that if α and β are orthogonal then σ_{α}σ_{β}σ_{α}^{−1} = σ_{β}^{±1} depending on whether α + β is a root or not (see the next section). The Lemma follows readily from this.

Recall (9.11)(f) W acts on X, and we refer to this as the cross action.

Lemma 14.17 Suppose α is imaginary with respect to x. Then wα is imaginary with respect to w × x and

(14.18)(a)  gr_{w×x}(wα) = gr_{x}(α).

Suppose gr_{x}(α) = 1. Then

(14.18)(b)  c^{wα}(w × x) = w × c^{α}(x).

We leave the elementary proof, and the statement of the corresponding facts for real roots, to the reader. See [27], Lemmas 4.15 and 7.11.

With Cayley transforms in hand we can complete the proof of Proposition 12.12. Proof of Propositions 12.12 and 12.18. Fix τ ∈ I_{W}. Assume there is a τ-imaginary root α. By Lemma 14.11(5) there exists x ∈ X_{τ}(α), so c^{α}(x) ∈ X_{s_{α}τ} is defined. Now suppose β is an imaginary root with respect to s_{α}τ. By the same argument we may choose x′ ∈ X_{s_{α}τ} so that x′′ = c_{β}(x′) is defined.

Replacing x ∈ X_{τ} with c_{α}(x′) ∈ X_{τ} we now have X_{τ} ∋ x → c^{β}c^{α}x ∈ X_{s_{β}s_{α}τ}. By Lemma 14.2(2) c^{β}c^{α}(x) is G-conjugate to x.

Continue in this way until we obtain x ∈ X_{τ}, x′ ∈ X_{τ′}, where x′ is Gconjugate to x, and there are no imaginary roots with respect to τ′. (This corresponds to the most split Cartan subgroup of the quasisplit form of G.) By [4, Proposition 6.24] θ_{ξ′} is quasisplit, for any ξ′ ∈ X̃ lying over x′.

This completes the proof of Proposition 12.12. The proof of Proposition 12.18 is similar, using inverse Cayley transforms, Proposition 12.16 and (12.17). We omit the details. □

Example 14.19 We conclude this section with some details in the case of G = Sp(4) (of type C_{2}). We give a picture of the space X and describe the action of Cayley transforms on X.

There are four G-conjugacy classes of strong involutions, which we think of as corresponding to Sp(2, 0), Sp(0, 2), Sp(1, 1) and the split group Sp(4, ℝ). See Example 5.9. Write (9.10) as

(14.20)  X = X_{2,0} ∪ X_{0,2} ∪ X_{1,1} ∪ X_{s}.

There are 4 conjugacy classes in I_{W}, corresponding to the 4 Cartan subgroups of Sp(4, ℝ), isomorphic to S^{1} × S^{1}, S^{1} × ℝ^{×}, ℂ^{×} and ℝ^{×} × ℝ^{×}.

Let α_{1}, α_{2} be the long positive roots, and β_{1}, β_{2} the short ones. Then I_{W} = {1, s_{α_{1}}, s_{α_{2}}, s_{β_{1}}, s_{β_{2}}, w_{0}} where w_{0} = −I is the long element.

Here is the output of the kgb command of the atlas software for the real form Sp(4, ℝ):

 0:  0  0  [n,n]    1   2     6   4
 1:  0  0  [n,n]    0   3     6   5
 2:  0  0  [c,n]    2   0     *   4
 3:  0  0  [c,n]    3   1     *   5
 4:  1  2  [C,r]    8   4     *   *  2
 5:  1  2  [C,r]    9   5     *   *  2
 6:  1  1  [r,C]    6   7     *   *  1
 7:  2  1  [n,C]    7   6    10   *  2,1,2
 8:  2  2  [C,n]    4   9     *  10  1,2,1
 9:  2  2  [C,n]    5   8     *  10  1,2,1
10:  3  3  [r,r]   10  10     *   *  1,2,1,2

Thus X_{s} has 11 elements, labelled by the first column. Each row corresponds to an orbit O, which maps to a twisted involution τ. In this case (since γ = 1) we may view τ as an involution in W; the last column gives τ as a product of simple reflections. The conjugacy class of τ corresponds to a Cartan subgroup; the number of this Cartan subgroup, given by the output of the cartan command, is given in column 3. The length of O, i.e. dim(O) − dim(O_{min}) where O_{min} is a minimal orbit, is given in column 2.

The type of each simple root (r=real, c=compact imaginary, n=noncompact imaginary, C=complex) is given in brackets. Following this the next two columns give the cross actions of the simple roots, followed by give Cayley transforms by the simple noncompact imaginary roots.

Of course X_{2,0} and X_{0,2} are singletons. Finally here is the output of kgb for Sp(1, 1).

0:  0  0  [n,c]   1  0    2  *
1:  0  0  [n,c]   0  1    2  *
2:  1  1  [r,C]   2  3    *  *  1
3:  2  1  [c,C]   3  2    *  *  2,1,2

Figure 2 gives a picture of X ^{3}. As before the vertical columns are the fibers X_{τ}. The numbering of the points of X is from the output of the kgb commands above. The arrows → indicate Cayley transforms.

[figure: •0′′′; Sp(2, 0); } ; |; ⟶ z = I; •0′′; Sp(0, 2); •0′; 1; { ; ; Sp(1, 1); •1′; 1; •2′; •3′; •0; 1; 2; { ; |; Sp(4, ℝ); } ; |; ⟶ z = −I; •1; 1; 2; •2; 2; •9; •5; •3; 2; •6; •7; 1; •8; 2; •4; 2; •10; 1; I_{W} :; s_{α_{1}}; s_{α_{2}}; s_{β_{1}}; s_{β_{2}}; −1; (S^{1})^{2}; Cartan:; ℂ^{×}; S^{1} × ℝ^{×}; (ℝ^{×})^{2}]

Figure 2: X and Cayley transforms for Sp(4)

See [2] for more detail about the representation theory of real forms of Sp(4, ℂ).

3Thanks to Leslie Saper for producing this diagram.

15 The Tits group and the algorithmic enumeration of parameters

The combinatorial enumeration of K\G/B is in terms of the Tits group W̃, introduced by Jacques Tits in [25] with the name extended Coxeter group.

We begin by fixing (G, γ) and a choice of of splitting datum S = (H, B, {X_{α}}) (cf. Section 2). For each simple root α let ϕ_{α} : SL(2) → G be defined by

ϕ_{α}(diag(z, \frac{1}{z})) = α^{∨}(z) and dϕ_{α} (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) = X_{α}. Let σ_{α} = ϕ_{α} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}). This

is consistent with the definition of σ_{α} in Section 14.

Definition 15.1 The Tits group W̃ is the subgroup of N generated by {σ_{α}} for α simple.

For each simple root α let m_{α} = σ_{α}^{2} = α^{∨}(−1). Let H_{0} be the subgroup of H generated by the elements m_{α}.

Theorem 15.2 (Tits [25])

(1) The kernel of the natural map W̃ → W is H_{0},

(2) The elements σ_{α} satisfy the braid relations,

(3) There is a canonical lifting of W to a subset of W̃: take a reduced expression w = s_{α_{1}} … s_{α_{n}}, and let w̃ = σ_{α_{1}} … σ_{α_{n}}.

It is a remarkable fact that the computations needed for the enumeration of the K-orbits on G/B can be carried out in the Tits group. This is due to the fact that Cayley transforms and cross actions can be described entirely in terms of the σ_{α}.

Lemma 15.3 Given x ∈ X, there exists a pre-image ξ of x in X̃ such that ξ normalizes W̃.

Proof. Recall (before Lemma 9.12) we write δ ∈ I_{W} for the image of the distinguished element δ of X. Using the fact that p(x) = wδ for some w ∈ W, it is easy to reduce to the case x ∈ X_{δ}. Let θ = int(δ) ∈ Aut(H).

Since δ is an automorphism of the based root datum used to define W̃, it follows easily that δσ_{α}δ^{−1} = σ_{θ(α)} for all roots α, and therefore δ normalizes W̃. Choose ξ ∈ X̃ mapping to x, so ξ = hδ for some h ∈ H; we have hθ(h) = ξ^{2} ∈ Z(G).

Recall (Remark 11.3) H = T_{τ}A_{τ}. We may replace ξ with gξg^{−1} = gθ(g^{−1})hδ for any g ∈ H. The map g ↦ gθ(g^{−1}) has image A_{τ}, so we may assume h ∈ T_{τ}.

We have ξσ_{α}ξ^{−1} = (hδ)σ_{α}(hδ)^{−1} = hσ_{β}(h^{−1})σ_{β} where β = θ(α). Now hθ(h) = h^{2} ∈ Z(G), so β(h) = ±1. It follows easily that hσ_{β}(h^{−1}) is equal to 1 or m_{β}, and is therefore contained in W̃. □

Fix x ∈ X, and choose a pre-image ξ ∈ X̃. We now describe an algorithm for computing X [x] ≃ K_{ξ}\G/B (cf. (9.8)). By Proposition 12.19(2) we may assume x ∈ X_{δ}, and by Lemma 15.3 that ξ normalizes W̃. For w̃ ∈ W̃ let θ(w̃) = ξw̃ξ^{−1}.

We will maintain an intermediate first-in-first-out list of pairs (τ, w̃) where τ ∈ I_{W}, w̃ ∈ W̃, and w̃ξ ∈ X̃_{τ}. We also maintain a store of elements W̃. If w̃ is in the store then the image of w̃ξ in X is contained in X [x]; denote this element w̃x. Initialize the list with (δ, 1).

For each (τ, w̃) occuring we will keep a record of of gr_{w̃x}; we assume we are given gr_{x}. Let M_{τ} be the subgroup of H_{0} generated by {m_{α} | τ(α) = α}. For each τ occuring we will compute M_{τ} ∩ A_{τ} (cf. (11.4)). We assume we are are given M_{δ} ∩ A_{δ}.

If the list is non-empty, remove from it the first element (τ, w̃).

First add w̃ to the store. At the first step (w̃ = 1), record gr_{x}. Otherwise w̃ is either of the form σ_{α}ũθ(σ_{α})^{−1} or σ_{α}ũ for some ũ already in the store (see below). Compute gr_{w̃x} by Lemmas 14.13 and 14.17, and record this information.

Next, compute the orbit of w̃x under the cross action of W_{i,τ} as follows. Suppose τ(α) = α and gr_{w̃x}(α) = 1. Choose a representative σ_{α} ∈ W̃ of s_{α} (α is not necessarily simple). By (14.16) σ_{α}w̃ξσ_{α}^{−1} = σ_{α}^{2}w̃ξ. Repeating this we obtain a collection of elements of the form {tw̃ξ | t ∈ S} where S is a subset of M_{τ}. The W_{i,τ}-orbit of x is in bijection with the image of S in the group M_{τ}/M_{τ} ∩ A_{τ}. Choose representatives of this set.

For each such representative t let ũ = tw̃ and add ũ to the store. Record gr_{ũx} by writing ũx = w × x for w ∈ W_{i,τ} and using Lemma 14.17. For each simple root α satisfying τ(α) = α and gr_{ũx}(α) = 1, see if any element of the form (s_{α}τ, ∗) is on the list. If not, add (s_{α}τ, σ_{α}ũ) to the list. Compute the set M_{s_{α}τ} ∩ A_{s_{α}τ} = ⟨M_{τ} ∩ A_{τ}, m_{α}⟩ (cf. Lemma 14.11).

Next, for each simple root α, check if (s_{α}τs_{α}, ∗) is on the list. If not add (s_{α}τs_{α}, σ_{α}w̃θ(σ_{α})^{−1}) to the list and compute M_{s_{α}τs_{α}} ∩ A_{s_{α}τs_{α}} = s_{α} · M_{τ} ∩ A_{τ}.

Continue until the list is empty, at which point X [x] is the set of elements w̃x for w̃ in the store.

Remark 15.4 Fix x ∈ X_{δ} and choose ξ as in the Lemma. The argument shows that the following information suffices to compute X [x]: W̃, the involution θ = int(ξ) of W̃, the grading gr_{x} of the δ-imaginary roots, and the two-group M_{δ} ∩ A_{δ} ⊂ W̃.

Note that if (the derived group of) G is simply connected then the grading gr_{x} is determined by W̃ and θ (cf. Remark 14.15).

Remark 15.5 We may also compute the entire space X by similar (in fact somewhat easier) methods, using Proposition 11.2 to describe the fiber X_{δ}, and Lemma 14.11 to compute Cayley transforms. See Remark 14.12. We omit the details.

Carrying out the algorithm uses computations in the Tits group, and we briefly sketch how to carry these out. According to Theorem 15.2, each element of the Tits group can be written uniquely as w̃h, with w̃ the canonical representative of w ∈ W and h ∈ H_{0}.

Fix w ∈ W and a simple root α, with corresponding reflection s = s_{α}. We first compute w̃s̃. If l(ws) > l(w), then w̃s̃ is the canonical representative of ws, and we have w̃s̃ = w̃s. Otherwise w = vs for v ∈ W with l(v) = l(w)−1. In this case w̃s̃ = ṽs̃^{2} = ṽm_{α}. There is a similar formula for s̃w̃.

Now for h ∈ H_{0} we have

(15.6)  (w̃h)s̃ = (w̃s̃)h^{s}, s̃(w̃h) = (s̃w̃)h

with w̃s̃ or s̃w̃ computed as above.

In addition, for h_{i} ∈ H_{0} we have

(15.7)  h_{1}(w̃h_{2})h_{3} = w̃(h_{1}^{w}h_{2}h_{3}).

Therefore multiplication in W̃ can be computed from multiplication in W, multiplication in H_{0}, and the action of W on H_{0}.

References

[1] Jeffrey Adams. Lifting of characters. In Harmonic analysis on reductive groups (Brunswick, ME, 1989), volume 101 of Progr. Math., pages 1–50. Birkhäuser Boston, Boston, MA, 1991.

[2] Jeffrey Adams. Guide to the atlas software: Computational representation theory of real reductive groups. In Representation Theory of Real Reductive Groups, Proceedings of Conference at Snowbird, July 2006, Contemp. Math. Amer. Math. Soc., 2008.

[3] Jeffrey Adams, Dan Barbasch, and David A. Vogan, Jr. The Langlands classification and irreducible characters for real reductive groups, volume 104 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1992.

[4] Jeffrey Adams and David A. Vogan, Jr. L-groups, projective representations, and the Langlands classification. Amer. J. Math., 114(1):45–138, 1992.

[5] A. A. Beĭlinson, J. Bernstein, and P. Deligne. Faisceaux pervers. In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pages 5–171. Soc. Math. France, Paris, 1982.

[6] A. Borel and H. Jacquet. Automorphic forms and automorphic representations. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 189–207. Amer. Math. Soc., Providence, R.I., 1979. With a supplement “On the notion of an automorphic representation” by R. P. Langlands.

[7] N. Bourbaki. Groupes et Algèbres de Lie, volume Chapitres 4,5 and 6 of Éléments de Mathématique. Masson, New York, 1981.

[8] P. Deligne and G. Lusztig. Representations of reductive groups over finite fields. Ann. of Math. (2), 103(1):103–161, 1976.

[9] Fokko du Cloux. Combinatorics for the representation theory of reduc-

tive groups. 2006. www.liegroups.org/papers/combinatorics.pdf.

[10] Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original.

[11] James E. Humphreys. Linear algebraic groups. Springer-Verlag, New York, 1975. Graduate Texts in Mathematics, No. 21.

[12] A. W. Knapp. Weyl group of a cuspidal parabolic. Ann. Sci. École Norm. Sup. (4), 8(2):275–294, 1975.

[13] Anthony Knapp. Representation Theory of Semisimple Groups. An overview based on. Princeton University Press, Princeton, NJ, 1986.

[14] Bertram Kostant. On the conjugacy of real Cartan subalgebras. I. Proc. Nat. Acad. Sci. U. S. A., 41:967–970, 1955.

[15] Bertram Kostant. On the conjugacy of real Cartan subalgebras. II. 1955. Preprint, to appear in Kostant’s collected works.

[16] R. P. Langlands. On the classification of irreducible representations of real algebraic groups. In Representation theory and harmonic analysis on semisimple Lie groups, volume 31 of Math. Surveys Monogr., pages 101–170. Amer. Math. Soc., Providence, RI, 1989.

[17] Toshihiko Matsuki. The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan, 31(2):331– 357, 1979.

[18] Dragan Miličić. Algebraic D-modules and representation theory of semisimple Lie groups. In The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992), volume 154 of Contemp. Math., pages 133–168. Amer. Math. Soc., Providence, RI, 1993.

[19] A. L. Onishchik and È. B. Vinberg. Lie groups and algebraic groups. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites.

[20] Vladimir Platonov and Andrei Rapinchuk. Algebraic groups and number theory, volume 139 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen.

[21] R. W. Richardson and T. A. Springer. The Bruhat order on symmetric varieties. Geom. Dedicata, 35(1-3):389–436, 1990.

[22] W. Schmid. On the characters of the discrete series. Inventiones Mathematicae, 30:47–144, 1975.

[23] T. A. Springer. Linear algebraic groups, volume 9 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, second edition, 1998.

[24] Mitsuo Sugiura. Conjugate classes of Cartan subalgebras in real semisimple Lie algebras. J. Math. Soc. Japan, 11:374–434, 1959.

[25] J. Tits. Normalisateurs de tores. I. Groupes de Coxeter étendus. J. Algebra, 4:96–116, 1966.

[26] David A. Vogan, Jr. Representations of Real Reductive Lie Groups, volume 15 of Progress in mathematics. Birkhäuser, Boston, 1981.

[27] David A. Vogan, Jr. Irreducible characters of semisimple Lie groups IV. character-multiplicity duality. Duke Math. J., 49, No. 4:943–1073, 1982.

[28] Gregg Zuckerman. Tensor products of finite and infinite dimensional representations of semisimple Lie groups. Ann. Math. (2), 106(2):295– 308, 1977.