Some notes on parametrizing representations

December 13, 2005

These notes are an attempt to clarify some confusing issues about parametrizing representations. This includes the basepoint issue. At least at the moment this is only of interest to the experts.

1 The crux of the biscuit

These notes have gotten rambling and disorganized, so here is the long and the short of it. For details see the relevant sections.

We want to go between Fokko’s data and some other parametrization of representations of G. The most direct route is to the modules I(Ψ^{+}, Λ) of [6].

The nice thing is that in the setting of [4] we fix almost everything, including H, B and even λ, the differential of Λ. The only thing which takes some work is specifying Λ off of the identity component of H(ℝ).

The data (y, λ) parametrize maps of W_{ℝ} into an E-group for a torus, but an additional choice is necessary (the distinguished element of this E-group) to actually get a representation of (a cover of) a torus. This is the basepoint question.

Many choices of basepoints will give a bijection between Fokko’s data and representations of strong real forms of G. These choices are pinned down by requiring that cross actions and Cayley transforms of representations correspond to the natural cross action and Cayley transform defined in [4].

Note that compatibility with cross action includes the fact that the gradings/cogradings of imaginary and compact roots behave correctly.

Compatibility with cross action primarily says that the basepoints we choose, i.e. certain elements y_{0} on the dual side, should be large. Furthermore compatibility with Cayley transforms says they should be obtained from a fixed δ^{∨} by a particular series of Cayley transforms. We spell the choices out in Section 5. There is a conjecture there, verified by Fokko by computer, about this issue (Conjecture 5.19). The work to show these choices are correct is in Section 6.

The bottom line is this: always use our fixed B and its corresponding set Ψ^{+} of positive roots. For a positive system of real roots use Ψ_{R,x}^{+}, the roots in Ψ^{+} which are real with respect to θ_{x}. Finally use the basepoints provided by Fokko’s algorithm. (These last two choices are related, you can modify them both simultaneously by W_{im}). Then you’ll get the correct parametrization.

1.1 Some Objects

It might be useful to collect some objects here which are defined later.

Our group is G, and we have fixed H and B. The parameter set is Z, and (x, y) is a typical element of Z.

An element x defines a real form H_{x}(ℝ) of H. A typical character of a cover of H_{x}(ℝ) is written Λ. We parametrize characters Λ as

Λ(x, λ, ν)

(cf. Lemma 2.1).

We define the cross action

w × Λ = Λ ⊗ μ

with μ = wλ − λ (Definition 2.14). This may also be written

w × Λ(x, λ, ν) = Λ(x, λ + μ, ν + μ).

The Cayley transform is

c^{α}Λ(x, λ, ν) = {Λ(σ_{α}x, λ, ν), Λ(σ_{α}x, λ, ν + α)}

(Definition 2.19).

We parametrize characters Λ in terms of maps of W_{ℝ} into the dual group as

Λ[x, y, λ, y_{0}]

(3.4)

We then compute the cross and Cayley transforms in these coordinates:

w × Λ(x, y, λ, y_{0}) = Λ(x, y, wλ, y_{0})

(Lemma 3.7) and

c^{α}Λ(x, y, λ, y_{0}) = Λ(σ_{α}x, σ^{α}y, λ, σ^{α}y_{0})

(Lemma 3.12).

A standard module is I(x, Ψ_{R}^{+}, Λ) (Definition 6.2). Here x specifies the (strong) real form, Ψ_{R}^{+} is a set of positive real roots with respect to θ_{x}, and Λ is a genuine character of a cover of H_{x}(ℝ). We may write Ψ_{R,x}^{+} to indicate that the real roots are defined by x. The differential of Λ is λ, which is fixed, and is the infinitesimal character. Usually we can take Ψ_{R}^{+} = Ψ_{R}^{+}, and the standard module is

I(x, Ψ_{R}^{+}, Λ)

and (Ψ_{R}^{+}, Λ) are in good position (Definition 6.1).

The cross action is given by (cf. Remark 1.1)

w × I(x, Ψ_{R}^{+}, Λ) = I(x, Ψ_{R}^{+}, w^{−1} × Λ)

(Lemma 6.13) and Cayley transforms by

c^{α}I(x, Ψ_{R,x}^{+}, Λ) = I(σ_{α}x, Ψ_{R,σ_{α}x}^{+}, c^{α}Λ)

(Proposition 6.21).

Finally we get to the parametrization of [4], so a standard module is

I(x, y, λ)

(Definition 6.7), and the actions are

w × I(x, y, λ) = I(w · x, w · y, λ)

and

c_{α}I(x, y, λ) = I(σ_{α}x, σ^{α}y, λ).

Remark 1.1 The cross action is defined on parameters, not representations. The only action canonically defined on representations is coherent continuation. Hence, as David points out, the notation w × I(x, Ψ_{R}^{+}, Λ) is misleading (even though it appears in [5].) I don’t want to fix this now.

One point is we could define w × I(x, ψ_{R}^{+}, Λ) to be ± the coherent continuation action of w on this representation, modulo representations coming from less compact Cartan subgroups.

2 Covers of Tori

Suppose H is a complex algebraic torus and γ ∈ \frac{1}{2}X^{∗}(H). Then the two-fold cover H_{γ} is defined. Suppose θ is a holomorphic involution of H. Then H_{θγ} is defined. We assume θγ −γ ∈ X^{∗}(H), so that H_{θγ} is canonically isomorphic to H_{γ}. (In our applications we will have γ = ρ.) Then θ acts on H_{γ} and we can talk about (𝔥, H_{γ}^{θ})-modules. These correspond to representations of a real form of a two-fold cover H_{γ}(ℝ) of H. See [6, Proposition 5.8] and [7].

Lemma 2.1 Fix γ ∈ X^{∗}(H) ⊗ ℂ and θ satisfying θγ − γ ∈ X^{∗}(H). Write H_{γ}(ℝ) for the corresponding real form of H_{γ}. Suppose (λ, ν) satisfy λ ∈ X^{∗}(H) ⊗ ℂ, ν ∈ γ + X^{∗}(H) and

(2.2)(a)  λ + θλ = ν + θν.

Then we obtain a genuine H_{γ}(ℝ)-module Λ(λ, ν) with differential λ. Furthermore Λ(λ, ν) = Λ(λ′, ν′) if and only if λ = λ′ and

(2.2)(b)  ν′ − ν ∈ (1 − θ)X^{∗}(H).

If γ ∈ X^{∗}(H) we recover representations of H(ℝ) as in [6, Proposition 3.26].

Remark 2.3 The dual Cartan involution is θ^{∨} = −θ, and we can write the Lemma in terms of θ^{∨} instead: Λ(x, λ, ν) is defined if

(2.4)(a)  λ − θ^{∨}λ = ν − θ^{∨}ν

and two are equal if

(2.4)(b)  ν′ − ν ∈ (1 + θ^{∨})X^{∗}(H).

The group H_{γ}(ℝ) comes equipped with a genuine character γ. The character 2γ factors to H_{γ}(ℝ). Its absolute value has a unique positive square root, which we denote |γ|. Then γ/|γ| is naturally a genuine representation of H_{γ}(ℝ). [Check: it doesn’t matter, as in [6, Example 8.13], that 2γ is not real valued?]

Lemma 2.5 1. γ = Λ(γ, γ).

2. |γ| = Λ(\frac{1}{2}(γ + \overline{γ}), 0).

3. γ/|γ| = Λ(\frac{1}{2}(γ − \overline{γ}), γ) Remark 2.6 In most applications we will have \overline{γ} = −θγ (since 2γ is in the root lattice) so these become:

1. γ = Λ(γ, γ).

2. |γ| = Λ(\frac{1}{2}(γ − θγ), 0).

3. γ/|γ| = Λ(\frac{1}{2}(γ + θγ), γ) In fact we’ll often have \overline{γ} = γ, i.e. θγ = −θγ, in which case these become

1. γ = Λ(γ, γ).

2. |γ| = Λ(γ, 0).

3. γ/|γ| = Λ(0, γ) Remark 2.7 Check these assertions.

For later use we do a few calculations in these coordinates. We suppose H is a θ-stable Cartan subgroup of G.

First of all if α is a root then

α = Λ(α, α)

and if α is real then sgn(α) = α/|α| and so

sgn(α) = Λ(0, α).

Write Ψ for the roots of H in G, and Ψ_{R} for the real roots. Suppose Ψ_{R}^{+} is a set of positive real roots and let ρ_{R} = \frac{1}{2} ∑_{α∈Ψ_{R}^{+}} α.

Note that θ(ρ_{R}) = −ρ_{R}, and 2ρ_{R} is real valued. Take γ = ρ_{R} so we have

ρ_{R}/|ρ_{R}| = Λ(0, ρ_{R})

Now take w ∈ W_{R}, the Weyl group of Ψ_{R}. Recall [6, (8.26)(a)] τ(Ψ_{R}^{+}, w):

(2.8)  τ(Ψ_{R}^{+}, w) = (ρ_{R}/|ρ_{R}|)/(wρ_{R}/|wρ_{R}|)

That is

τ(Ψ_{R}^{+}, w) = Λ(0, ρ_{R} − wρ_{R})

In particular if α is a simple root of Ψ_{R}^{+} we have

(2.9)  τ(Ψ_{R}^{+}, s_{α}) = Λ(0, α) = sgn(α).

We may drop Ψ_{R}^{+} from the notation.

2.1 Cross action on Λ

Now suppose H is a Cartan subgroup of G. A critical role is played by the action of W (G, H). Suppose x is a strong real form of G, let θ = θ_{x} and assumem H is θ stable. Also assume θγ −γ ∈ X^{∗}(H). We talk about (𝔥, H_{γ}^{θ}) modules, but we should also keep the strong real form x of H in mind. So we write Λ(x, λ, ν) to indicate this is a representation of the strong real form x of H.

Suppose w ∈ W = W (G, H). Then w acts on everything in sight, and

Definition 2.10 Suppose Λ = Λ(x, λ, ν). Then

(2.11)  wΛ = Λ(wx, wλ, wν)

This is only a representation of the same group if wx = x, i.e. w ∈ W^{θ}. In this case we have:

Lemma 2.12 Suppose w ∈ W^{θ}, i.e. wx = x. Then w acts on H_{γ}(ℝ), therefore on genuine characters of this group, and the action just described agrees with this action.

Remark 2.13 Check this - it must be true…

We also define the cross action of W. Suppose Λ is a character of H_{γ}(ℝ) and w ∈ W (dΛ), i.e. w(dΛ) = dΛ + μ for μ ∈ X^{∗}(H).

Definition 2.14 Let

w × Λ = Λ ⊗ μ

See [5, page 44] and [8, Definition 8.3.1]. This definition is simpler than the of [8], because it is in terms of Λ rather then Γ.

Lemma 2.15 Write wλ = λ + μ with μ ∈ X^{∗}(H).

w × Λ(x, λ, ν) = Λ(x, λ + μ, ν + μ)
= Λ(x, wλ, ν + μ)
= Λ(x, wλ, ν + (wλ − λ))

In particular

w × Λ(x, λ, λ) = Λ(x, wλ, wλ)

We emphasize the difference between wΛ and w × Λ:

\begin{aligned}wΛ(x, λ, ν) & = Λ(wx, wλ, wν) \\ w × Λ(x, λ, ν) & = Λ(x, wλ, ν + (wλ − λ). \end{aligned}

In particular

w^{−1}(w × Λ(x, λ, ν)) = Λ(w^{−1}x, λ, w^{−1}(ν + μ))
= Λ(w^{−1}x, λ, w^{−1}ν + (λ − w^{−1}λ))
= Λ(w^{−1}x, λ, ν + (λ − ν) − w^{−1}(λ − ν))

This latter operation is relevant since it gives us a character with the same differential, albeit possibly for a different group.

Finally suppose α is a root and set m = ⟨λ, α^{∨}⟩, n = ⟨ν, α^{∨}⟩. Then

(2.16)  \begin{aligned}s_{α}Λ(x, λ, ν) & = Λ(s_{α}x, λ − mα, ν − nα) \\ s_{α} × Λ(x, λ, ν) & = Λ(x, λ − mα, ν − mα) \\ s_{α}(s_{α} × Λ(x, λ, ν)) & = Λ(s_{α}x, λ, ν + (m − n)α) \end{aligned}

An important special case is w = s_{α} for α a real root:

(2.17)  s_{α}(s_{α} × Λ(x, λ, ν)) = Λ(x, λ, ν + (λ − ν) − s_{α}(λ − ν))

Recall τ(s_{α}) = Λ(0, ρ_{R} − s_{α}ρ_{R}). For later use we note

(2.18)

s_{α}(s_{α} × Λ(x, λ, ν)) ⊗ τ(s_{α}) = Λ(x, λ, ν + (λ − ν + ρ_{R}) − s_{α}(λ − ν + ρ_{R}))
= Λ(x, λ, ν) ⊗ Λ(0, ⟨λ − ν + ρ_{R}, α^{∨}⟩α)
= Λ(x, λ, ν) ⊗ sgn(α)^{⟨λ−ν+ρ_{R},α^{∨}⟩}

2.2 Cayley transform of Λ

We now turn to Cayley transforms.

Suppose Λ = Λ(x, λ, ν), of course (λ, ν) satisfy (2.2)(a) and θ = θ_{x}. Suppose α is a non-compact imaginary root for θ.

Definition 2.19

(2.20)  c^{α}Λ(x, λ, ν) = {Λ(σ_{α}x, λ, ν), Λ(σ_{α}x, λ, ν + α)}

Here σ_{α} is as in [4]. We check the characters on the right hand side are well defined. We have θ_{c_{α}x} = s_{α}θ.

(λ + s_{α}θλ) − (ν + s_{α}θν) = (λ + θλ − ⟨θλ, α^{∨}⟩) − (ν + θν − ⟨θν, α^{∨}⟩α
= (λ + θλ) − (ν + θν) + ⟨θ(ν − λ), α^{∨}⟩
= ⟨ν − λ, α^{∨}⟩ by (2.2)(a)

We need to show this is 0. By (2.2)(a) we have

⟨(λ − ν) + (θλ − θν), α^{∨}⟩ = 0

and since θα^{∨} = α^{∨} we conclude

2⟨λ − ν, α^{∨}⟩ = 0

Note that the two characters on the right hand side of (2.20) might be same. Note that

Λ(σ_{α}x, λ, ν + α) = Λ(σ_{α}x, λ, ν) ⊗ sgn(α)

and these are different if α : H_{c_{α}x}(ℝ) → ℝ^{∗} is surjective, i.e. α (now thought of as a real root) is type I. Equivalently by (2.2)(b) this holds if

α ∈ (1 − θ)X^{∗}(H).

For example if G = SL(2) and α is a real root then α = (1 − θ)α/2, so these are the same. If G = P GL(2) then α/2 ∉ X^{∗}(H) and these characters are distinct.

3 Maps of the Weil group

Fix y. The data (y, λ) defines a map ϕ : W_{ℝ} → ⟨H^{∨}, y⟩. Choose y_{0} ∈ H^{∨}y, i.e. p(y_{0}) = p(y). Then ⟨H^{∨}, y⟩ = ⟨H^{∨}, y_{0}⟩ is an E-group for H; by this we mean we consider y_{0} to be the distinguished element [6, Definition 5.9]. Recall

(3.1)(a)  y^{2} = e^{2πiλ}

and λ ∈ X^{∗}(H) since we are in the integral case. Choose γ so that

(3.1)(b)  y_{0}^{2} = e^{2πiγ}

and recall 2γ ∈ X^{∗}(H). Write

(3.1)(c)  y = e^{2πiτ}y_{0}

for some τ ∈ X^{∗}(H^{∨}) ⊗ ℂ. Squaring both sides and using (3.1)(a,b) we conclude

(3.1)(d)  e^{2πiγ} = e^{2πi(λ−(τ+θ^{∨}τ))}

i.e.

(3.1)(e)  λ − (τ + θ^{∨}τ) ∈ γ + X^{∗}(H)

Definition 3.2 Given (y, λ, y_{0}) let ϕ(y, λ, y_{0}) : W_{ℝ} → ⟨H^{∨}, y_{0}⟩ be the corresponding map. That is

(3.3)  \begin{aligned}ϕ(z) & = z^{λ}\overline{z}^{yλ} \\ ϕ(j) & = e^{−πiλ}y \\ & = e^{2πi(τ− \frac{1}{2} λ)}y_{0} \end{aligned}

Suppose (x, y) ∈ Z. Then ϕ defines a genuine character

(3.4)  Λ[x, y, λ, y_{0}]

of H_{γ}(ℝ).

Lemma 3.5 Suppose we are given (x, y) ∈ Z and (λ, y_{0}). We assume e^{2πiλ} = y^{2} as usual. Choose τ satisfying

y = e^{2πiτ}y_{0}.

Then in the notion of Section 2 we have

Λ[x, y, λ, y_{0}] = Λ(x, λ, κ)

with

κ = λ − (τ + θ^{∨}τ)

This is a representation of H_{γ}(ℝ) where e^{2πiγ} = y_{0}^{2}.

Note that by (3.1)(e) κ ∈ γ + X^{∗}(H) so this defines a representation of the γ cover of H_{x}(ℝ).

Example 3.6 For example suppose y = y_{0}, so h = 1, λ = γ and τ = 0. Then

Λ = χ(x, λ, λ).

This is a representation of H_{x}(ℝ)_{λ}, which is the trivial cover if λ ∈ X^{∗}(H). This is (the restriction of) a holomorphic character of H_{λ}.

Proof. The main point is that

ϕ(z) = z^{λ}\overline{z}^{yλ}
ϕ(j) = e^{−πiλ}y
= e^{−πiλ}hy_{0}
= e^{−πiλ+2πiτ}y_{0}
= e^{2πi(τ− \frac{1}{2} λ)}y_{0}

Then by [6, (4.7)(c)] Λ = Λ(x, λ, κ) with

\begin{aligned}κ & = \frac{1}{2}(λ − θ^{∨}λ) − ((τ − \frac{1}{2}λ) + θ^{∨}(τ − \frac{1}{2}λ)) \\ & = λ − (τ + θ^{∨}τ) \end{aligned}

□

3.1 The Cross Action

We now have to compute the standard and cross actions of W in terms of the parameters Λ[x, y, λ, y_{0}]. Lemma 3.7

(3.8)  w × Λ[x, y, λ, y_{0}] = Λ[x, y, wλ, y_{0}]

Proof. This is perhaps surprising. Let’s see, write y = e^{2πiτ}y_{0}, and then

(3.9)  \begin{aligned}w × Λ[x, y, λ, y_{0}] & = w × Λ(x, λ, λ − (τ + θ^{∨}τ)) \\ & = Λ(x, wλ, λ − (τ + θ^{∨}τ) + (wλ − λ)) \\ & = Λ(x, wλ, wλ − (τ + θ^{∨}τ)) \\ & = Λ[x, y, wλ, y_{0}] \end{aligned}
□

Lemma 3.10

(3.11)  wΛ[x, y, λ, y_{0}] = Λ[wx, wy, wλ, wy_{0}]

This is the natural action of W.

We sometimes combine the two:

w^{−1}(w × Λ[x, y, λ, y_{0}]) = Λ[w^{−1}x, w^{−1}y, λ, w^{−1}y_{0}]

which is handy since this has the same λ.

3.2 Cayley Transforms

We next need to compute c^{α} in the Λ[x, y, λ, y_{0}] coordinates. Lemma 3.12 Suppose α is a noncompact imaginary root. Then

(3.13)  c^{α}Λ[x, y, λ, y_{0}] = Λ[σ_{α}x, σ^{α}y, λ, σ^{α}y_{0}]

Here σ^{α} is the possibly multivalued inverse cayley transform. That is σ^{α}y = {y′ | σ_{α}y′ = y}.

Proof. Let θ = θ_{x}. Write y = e^{2πiτ}y_{0}, so

Λ[x, y, λ, y_{0}] = Λ(x, λ, ν)

with

ν = λ − (τ + θ^{∨}τ).

Then

\begin{aligned}c^{α}Λ[x, y, λ, y_{0}] & = c^{α}Λ(x, λ, ν) \\ & = Λ(σ_{α}x, λ, ν) \end{aligned}

On the other hand we have

\begin{aligned}σ_{α}y & = σ_{α}e^{2πiτ}y_{0} \\ & = e^{2πis_{α}τ}σ_{α}y_{0} \end{aligned}

so

Λ[σ_{α}x, σ^{α}y, λ, σ^{α}y_{0}] = Λ(σ_{α}x, λ, ν′)

with

\begin{aligned}ν′ & = λ − (s_{α}τ + s_{α}θ^{∨}s_{α}τ) \\ & = λ − (s_{α}τ + θ^{∨}τ) \end{aligned}

Therefore

ν′ − ν = λ − (s_{α}τ + θ^{∨}τ) − λ + (τ + θ^{∨}τ)
= τ − s_{α}τ
= ⟨τ, α^{∨}⟩α^{∨}

We’re on the less compact Cartan here, i.e. α is a real root here, or in other words s_{α}θα = −α. Therefore α^{∨} = (1 − s_{α}θ)\frac{1}{2}α^{∨}, so the result follows from (2.2)(b).

□

4 The Setup

We need to go back and forth a bit between the notation of Fokko’s notes [4] and that of [3]. See also [1], [5].

As in [3, Section 8] we are given (G, S). Here G is a (complex) reductive algebraic group, and S is a conjugacy class of (distinguished) splittings of the usual exact sequence, in other words a conjugacy class of elements δ such that θ_{δ} is a fundamental involution. To avoid confusion we may write such elements as δ_{f}.

Equivalently we are given a conjugacy class D of pairs (δ, B) where B is large with respect to δ. Thus int(δ) is a quasisplit involution, so we may write it δ_{q}. Recall [6, 9.7(d)]

δ_{q} = m_{ρ}δ_{f}

Similarly on the dual side we have G^{∨}, δ_{f}^{∨} ∈ S^{∨}, and (δ_{q}^{∨}, B^{∨}) ∈ D^{∨}. We will work in the Fokko’s setting, so we will fix δ, δ^{∨} and H, B, H^{∨}, B^{∨}.

Here H is fundamental, and B is large, with respect to δ = δ_{f}, and similarly on the dual side.

Recall the definitions of X , I and Z [4]. In particular I ⊂ W δ is the set of twisted involutions, and there is a map p : X → I. We write σ· for the action of σ ∈ W on X and I. This map is equivariant for the actions of W on X and I.

Our basic data will be a pair (x, y) ∈ Z. Together with an element λ ∈ X_{∗}(H^{∨}) ⊗ ℂ, satisfying e^{2πiλ} = y^{2}, these will parametrize various things, including maps of Weil groups, characters of Cartans of various sorts, and representations. This is what we want to make precise.

An important but barely visible role is played by ζ. In the setting of [4] ζ is fixed, and gives identifications

X^{∗}(H) = X_{∗}(H^{∨}), X_{∗}(H) = X^{∗}(H^{∨}).

In particaulr if α ∈ X^{∗}(H) is a root of H in G then α^{∨} ∈ X_{∗}(H) = X^{∗}(H^{∨}) is a root of H^{∨} in G^{∨}. We use these implicitly. We also identify W (G, H) and W (G^{∨}, H^{∨}).

Our fixed Borel B gives a fixed set Ψ^{+} (note the bold face) of positive roots.

4.1 Some notation about root systems

The notation is verging on incomprehensible. This is an attempt to clarify it.

We have our fixed root system Ψ, and positive roots Ψ^{+}.

If x ∈ X then θ_{x} is an involution of G. Its restriction to H only depends on p(x) ∈ I. We write the real roots as Ψ_{R,x} or Ψ_{R,w} to indicate this dependence, and Ψ_{im,x}Ψ_{im,w} similarly.

We define Ψ_{R,x}^{+} = Ψ^{+} ∩ Ψ_{R,x}, and other similar versions.

If Ψ^{+} is an arbitrary set of positive roots, we define Ψ_{R,x}^{+} etc. the obvious way.

On occasion we will talk about simple roots for Ψ, and also for Ψ_{im,x}^{+}. In the latter case we mean a simple root for the root system Ψ_{im,x}, as opposed to a simple root for Ψ which happens to be in Ψ_{im,x}.

In Section 5 we introduce notation

Ψ_{im}^{+}[x].

This is a set of positive roots for Ψ_{im,x}; its dependence on x is subtle.

5 Choice of Basepoints

Recall [4] I ⊂ W δ is the set of twisted involutions, and p : X → I. We write u· for the action of u ∈ W on X and I. Also recall we have chosen an element δ so that B is large with respect to θ_{δ}.

We need a result of the following form. Recall X (w) = {x ∈ X | p(x) = w}. For w ∈ I let

(5.1)  X (w, δ) = {x ∈ X (w) | x is conjugate to δ}

This set is not empty [6, Lemma 9.17]. Recall the restriction of θ_{x} to H only depends on the image of x in I, and we write θ_{x} = θ_{w} accordingly. As usual let Ψ_{im,w} be the imaginary roots with respect to θ_{w} and let W_{im,w} = W (Ψ_{im,w}). Then X (w, δ) a single W_{im,w}-orbit.

For x ∈ X let

(5.2)  X (x, δ) = X (p(x), δ).

Fix θ. We say a set Ψ^{+} of positive roots is large if every simple root is complex or non-compact imaginary. In particular if Ψ_{im}^{+} is a set of positive imaginary roots, it is large if every simple root is non-compact.

Fix w. We want a W_{im,w} equivariant map which takes x ∈ X (w, δ) to a W (K_{x}, H)-orbit of large sets Ψ_{im,w}^{+}[x] with respect to θ_{x}.

Assumption 5.3 Given w ∈ I and x ∈ X (w, δ) we have chosen a set Ψ_{im,w}^{+}[x] of positive imaginary roots, large with respect to θ_{x}, and defined up to conjugacy by W (K_{x}, H). We assume these sets satisfy

(5.4)  Ψ_{im,u·w}^{+}[u · x] = uΨ_{im,w}^{+}[x] (u ∈ W)

(up to the action of W (K_{u·w}, H) on the left and W (K_{x}, H) on the right).

In particular

(5.5)  Ψ_{im,u·w}^{+}[u · x] = uΨ_{im,w}^{+}[x] (u ∈ W_{im}).

Given Ψ_{im,w}^{+}[x] for a single x ∈ X (w, δ) this defines Ψ_{im}^{+}[x] for all x′ ∈ X (w, δ).

A natural choice of Ψ_{im,w}^{+} is Ψ_{im,w}^{+}, where Ψ^{+} is our fixed set of positive roots (cf. Section 4). This is large with respect to some small number of choices of x ∈ X (w, δ). For example if w = p(δ) then we can take x = δ. There may be other choices.

For example if G = SL(2) and w = 1 we may take x = ±δ. Of course the natural choice is δ. Note that wδ = −δ, but w(δ, B) = (−δ, \overline{B}).

Condition (5.4) now takes the following form.

Assumption 5.6 For each w ∈ I we are given an element x_{b}[w] ∈ X (w, δ) (the “basepoint”) satisfying: Ψ_{im,w}^{+} is large with respect to x_{b}[w]. These satisfy the following condition. Suppose u ∈ W, and choose σ ∈ W satisfying

(5.7)(a)  x_{b}[u · w] = σ · x_{b}[w]

Then

(5.7)(b)  Ψ_{im,u·w}^{+} = σΨ_{im,w}^{+}.

Remark 5.8 For example if u · w = w, i.e. u ∈ W^{θ_{w}}, we may take σ = 1, and (5.7)(a) is immediate.

Remark 5.9 Note that u · x_{b}[w] ∈ X (u · w, δ), which implies σ = τu with τ ∈ W_{im,u·w}.

Definition 5.10 Given w let x_{b} = x_{b}[w] as in Assumption 5.6. Let

(5.11)(a)  Ψ_{im,w}^{+}[x_{b}] = Ψ_{im,w}^{+}

Define Ψ_{im,w}^{+}[x] for all x ∈ X (w, δ) by (5.5). That is, given x ∈ X (w, δ) choose σ ∈ W (Ψ_{im,w}) satisfying σ · x_{b} = x and define

(5.11)(b)  Ψ_{im,w}^{+}[x] = σΨ_{im,w}^{+}.

By (5.4) this defines Ψ_{im,w}^{+}[x] (up to conjugacy by W_{im}(K_{x}, H)) for all x ∈ X (w, δ).

We check that in fact Definition 5.10 implies (5.4) holds.

Lemma 5.12 Assume the basepoints x_{b}[w] satisfy Assumption 5.6. Then the map x → Ψ_{im,w}^{+}[x] of Definition 5.10 satisfies (5.4).

Proof. Fix w ∈ I, x ∈ X (w, δ), and u ∈ W. We have to show

(5.13)(a)  Ψ_{im,u·w}^{+}[u · x] = uΨ_{im,w}^{+}[x]

For the left hand side choose σ ∈ W (Ψ_{im,u·w}) so that

(5.13)(b)  u · x = σ · x_{b}[u · w].

Similarly choose τ ∈ W (Ψ_{im,w}) so that

(5.13)(c)  x = τ · x_{b}[w]

Note that these two relations imply

(5.13)(d)  x_{b}[u · w] = σ^{−1}uτ · x_{b}[w]

We have to show

(5.13)(e)  σΨ_{im,u·w}^{+} = uτΨ_{im,w}^{+}

i.e.

(5.13)(f)  Ψ_{im,u·w}^{+} = σ^{−1}uτΨ_{im,w}^{+}.

This follows from (5.13)(d) and Assumption 5.6. □

Fix w ∈ I and x_{b}[w]. Suppose α is a real or imaginary root with respect to θ_{w}. Then s_{α} · w = w so

x_{b}[s_{α} · w] = x_{b}[w].

(Warning: if α is real then x_{b}[s_{α} · w] = s_{α} · x_{b}[w] = x_{b}[w], but this may fail if α is non-compact imaginary, since such s_{α} may act non-trivially on X (w).)

Now suppose α is a simple root of Ψ^{+} which is complex. It is easy to see that

(5.14)  s_{α}Ψ_{im,w}^{+} = Ψ_{im,s_{α}·w}^{+}.

(The main point is that the imaginary the roots of Ψ^{+} and s_{α}Ψ^{+} are the same, since s_{α} only changes the sign of the complex root α). This shows that in (5.7)(a-b) if u = s_{α} we may take σ = s_{α}, i.e.

x_{b}[s_{α} · w] = s_{α} · x_{b}[w]

for α a complex simple root. This gives:

Lemma 5.15 Suppose that for all w ∈ I we are given x ∈ X (w, δ) such that Ψ_{im,w}^{+} is large with respect to x_{b}[w]. These choices satisfy the second condition of Assumption 5.6 if the following condition holds for all w ∈ I. Suppose α is a simple root for Ψ^{+} and is complex with respect to w. Then

(5.16)  x_{b}[s_{α} · w] = s_{α} · x_{b}[w]

So far these choices do two things: ensure that our definition of standard modules is independent of the choice of Ψ_{R}^{+} (Definition 6.7 and Proposition 6.9), and gives the correct cross action (Proposition 6.14).

We also need some compatibility of these choices under Cayley transforms.

Lemma 5.17 Suppose Ψ_{im,w}^{+} is large with respect to θ_{w}. Let α be an imaginary non-compact root which is simple for Ψ_{im,w}^{+}. Then Ψ_{im,σ_{α}w}^{+} is large with respect to σ_{α}w.

Proof. David sketched a proof of this. Becky Herb gave a complete case by case proof. □

This tells us we could choose x_{b}[σ_{α}w] = σ_{α}x_{b}[w] with α as in the Lemma. We put this all together.

Proposition 5.18 There exists at most one choice of {x_{b}[w] | w ∈ I} satisfying the following conditions for all w ∈ I. Define real, imaginary, complex, compact and non-compact with respect to θ_{w}.

1. x_{b}[p(δ)] = δ.

2. Ψ_{im,w}^{+} is large with respect to x_{b}[w].

3. Suppose α is a simple root of Ψ^{+}. Then

x_{b}[s_{α} · w] = \begin{cases}x_{b}[w] & if α is real or imaginary \\ s_{α} · x_{b}[w] & if α is complex \end{cases}

4. Suppose α is imaginary, non-compact, and simple for Ψ_{im,w}^{+}. Then

x_{b}[σ_{α}w] = σ_{α}x_{b}[w]

Here σ_{α} is an element mapping to s_{α} as in [4]. Recall this is the Cayley transform action, i.e. left multiplication by σ_{α}.

The proof is immediate; starting at p(δ) we may get to any w ∈ I by a series of Cayley transforms and cross actions as in the Proposition.

Conjecture 5.19 There exists x_{b}[w] as in the Proposition.

Fokko has built a check of this into the atlas software.

We will apply this on the dual side.

6 Standard Modules

We work in the setting of [4], so we have fixed H, B, H^{∨}, B^{∨}.

Definition 6.1 ([6], Definition 8.18) Fix θ, a set Ψ^{+} of positive roots, and an element λ ∈ X^{∗}(H) ⊗ ℂ. We say (Ψ^{+}, λ) is in good position if the following conditions hold.

1. If α is real then ⟨λ, α^{∨}⟩ < 0,

2. If α is imaginary then ⟨λ, α^{∨}⟩ > 0,

3. Suppose α, θα are positive complex roots. Then ⟨λ, α^{∨}⟩ > 0 or ⟨λ, θα^{∨}⟩ > 0.

4. Suppose α, −θα are positive complex roots. Then ⟨λ, α^{∨}⟩ < 0 or ⟨λ, θα^{∨}⟩ < 0.

Now suppose Ψ_{R}^{+} is the set of positive real roots. We say (Ψ_{R}^{+}, λ) is in good position if ⟨λ, α^{∨}⟩ < 0 for all α ∈ Ψ_{R}^{+}.

If Λ is a character we say (Ψ^{+}, Λ) is in good position if (Ψ^{+}, dΛ) is in good position, and similarly Ψ_{R}^{+}, Λ).

Definition 6.2 Fix x. Suppose Ψ_{R,x}^{+} is an arbitrary set of positive real roots and Λ is a genuine H_{ρ}(ℝ)-module. Let λ = dΛ. Then the standard module

I(x, Ψ_{R,x}^{+}, Λ)

is defined [6, Definition 8.27].

Since x is already specified we will sometimes drop it from the notation and write

I(x, Ψ_{R}^{+}, Λ).

We have included x to indicate the strong real form; this is a (𝔤, K_{x})-module.

We can always reduce to the case (Ψ_{R}^{+}, Λ) in good position by [6, Lemma 8.24]:

(6.3)  I(x, Ψ_{R}^{+}, Λ) ≃ I(x, wΨ_{R}^{+}, Λ ⊗ τ(Ψ_{R}^{+}, w))

for w ∈ W_{R}.

Recall τ(Ψ_{R}^{+}, w) is a ℤ/2ℤ-valued character of H_{x}(ℝ). See (2.8) and [6, Proposition 8.24].

Remark 6.4 In [6, Definition 8.27] there is also a choice of positive real roots P. Since λ is regular P is determined by λ and we have dropped it from the notation. While Ψ_{R}^{+} is also determined by λ if it is in good position, we do not want to assume good position, so we do not drop it from the notation.

It is worth noting that

(6.5)(a)  I(wx, Ψ_{R,wx}^{+}, Λ) ≃ I(x, wΨ_{R,x}^{+}, wΛ) (w ∈ W (G, H)).

That is there is an isomorphism from K_{x} to K_{wx} taking the (𝔤, K_{x})-module I(x, Ψ_{R}^{+}, Λ) to the (𝔤, K_{wx})-module I(wx, wΨ_{R}^{+}, wΛ). In particular

(6.5)(b)  I(x, Ψ_{R}, Λ) ≃ I(x, wΨ_{R}, wΛ) (w ∈ W (K_{x}, H)).

Lemma 6.6 Suppose (Ψ, Λ) and (Ψ, Λ′) are in good position. Then I(x, Ψ, Λ) ≃ I(x, Ψ, Λ′) if and only if Λ ≃ wΛ′ for some w ∈ W (K_{x}, H) satisfing wΨ_{R}^{+} = Ψ_{R}^{+}.

Note: I think this isn’t right - you might need to allow moving the positive system and a τ term - see [6].

Because of this we sometimes want to put our data in good position.

Fix an infinitesimal character (regular integral) and write it as λ for λ ∈ X_{∗}(H)⊗ℂ B^{∨}-dominant (identified with a B-dominant element of X^{∗}(H)⊗ℂ via our fixed ζ).

Now we give the crucial definition of the representation associated to L-data.

Definition 6.7 Suppose (x, y) ∈ Z and e^{2πiλ} = y^{2}. Choose y_{0} ∈ X (y, δ^{∨}). The set Ψ_{im,y}^{+}[y_{0}] for G^{∨} (Assumption 5.6) corresponds to a set Ψ_{R,x}^{+}[y_{0}] of positive real roots for G. Let Ψ_{im,y}^{−}[y_{0}] = −Ψ_{im,y}^{+}[y_{0}] (cf. Definition 6.1(1)).

Define

(6.8)(a)  I(x, y, λ) = I(x, Ψ_{R,x}^{−}[y_{0}], Λ[x, y, λ, y_{0}])

Let y_{b} = y_{b}[y], and recall Ψ_{im,y}^{+}[y_{b}] = Ψ_{R,y}^{+} (Assumption 5.6). By Proposition 6.9 we may choose y_{0} = y_{b}, and we see

(6.8)(b)  I(x, y, λ) = I(x, Ψ_{R,x}^{−}, Λ[x, y, λ, y_{b}])

This is a (𝔤, K_{x})-module, or more casually a G_{x}(ℝ)-module.

The next Lemma provides one of the justifications for Assumption 5.6.

Proposition 6.9 The module I(x, y, λ) is independent of the choice of y_{0} ∈ X (y, δ^{∨}).

Implicit in this is the assertion that the module is independent of the choice of Ψ_{R,x}^{−}[y_{0}], which is only defined up to conjugacy by K_{y}^{∨}. Proof. Let θ^{∨} = θ_{y}, w_{0}^{∨} = p(y) ∈ I. Suppose y′_{0} is another choice, i.e. y′_{0} = w^{∨}y_{0} for w^{∨} ∈ W^{θ^{∨}} (on the dual side). The action of this group on X (w_{0}^{∨}) is trivial, except for the action of W_{im}.

We need to show for w^{∨} ∈ W_{im}(G^{∨}, H^{∨}), with w = ζ^{−1}(w^{∨}) ∈ W_{R}(G, H)

I(x, Ψ_{R}^{−}(w^{∨}y_{0}), Λ(x, y, λ, w^{∨}y_{0})) = I(x, Ψ_{R}^{−}(y_{0}), Λ(x, y, λ, y_{0}))

By (5.4) and (6.3)

\begin{aligned}I(x, Ψ_{R}^{−}(w^{∨}y_{0}), Λ(x, y, λ, w^{∨}y_{0})) & = I(x, wΨ_{R}^{−}(y_{0}), Λ(x, y, λ, w^{∨}y_{0})) \\ & = I(x, Ψ_{R}^{−}(y_{0}), Λ(x, y, λ, w^{∨}y_{0}) ⊗ τ(w)) \end{aligned}

So it is enough to show

Lemma 6.10

(6.11)  Λ[x, y, λ, w^{∨}y_{0}] = Λ[x, y, λ, y_{0}] ⊗ τ(w)

This is basically [6, Lemma 9.28]. Proof. It is enough to check this with w = s_{α} for α a simple root of Ψ_{R}. Then α^{∨} is an imaginary root. As in Section 3 write

y = e^{2πiτ}y_{0}

so

Λ[x, y, λ, y_{0}] = Λ(x, λ, κ)

with

κ = λ − (1 + θ^{∨})τ

Recall (2.9)

τ(s_{α}) = Λ(0, α^{∨})

so the right hand side of (6.11) is

Λ(x, λ, κ′)

with

\begin{aligned}κ′ & = λ − (1 + θ^{∨})τ + α^{∨} \\ & = λ − (1 + θ^{∨})(τ + \frac{1}{2}α^{∨}) \end{aligned}

To compute Λ[x, y, λ, s_{α^{∨}}y_{0}] we write s_{α^{∨}}y_{0} = gθ^{∨}(g^{−1})y_{0} where g ∈ Norm_{G^{∨}}(H^{∨}) gives s_{α^{∨}}. Write gθ^{∨}(g^{−1}) = e^{2πiη}. Then

y = e^{2πi(τ+η)}y_{0}

and therefore

Λ[x, y, λ, s_{α^{∨}}y_{0}] = Λ(x, λ, κ′′)

where

κ′′ = λ − (1 + θ^{∨})(τ + η)

So it is enough to show

η = \frac{1}{2}α^{∨}

i.e.

gθ^{∨}(g^{−1}) = α^{∨}(−1) = m_{α}.

Now it is crucial that y_{0} is large, so that α^{∨} is non-compact imaginary. This is now an SL(2) calculation. See the last formula on page 126 of [6].

This proves the the Lemma. □

This completes the proof of the Proposition. □

Remark 6.12 In the terminology of Section 2 Λ = Λ(x, λ, ν) for this given λ, and some ν. The main thing to keep track of is therefore ν.

With this setup x hardly matters at all. Its only role is to define the real forms of G and H.

For example suppose G is connected and contains a compact Cartan subgroup, and suppose λ ∈ ρ + X^{∗}(H). The discrete series with infinitesimal character λ are the set of

I(x, ∅, Λ(x, λ, 0))

as x varies. Each Λ(x, λ, 0) is the “same” character of H; the only thing which changes is the real form x of H and G.

6.1 Cross Action

Suppose I(x, Ψ_{R}^{+}, Λ) is a standard module and w ∈ W = W (G, H). See Remark 1.1.

Lemma 6.13 ([5])

w × I(x, Ψ_{R}^{+}, Λ) ≃ I(x, Ψ_{R}^{+}, w^{−1} × Λ)

We need to compute this in I(x, y, λ) coordinates.

Proposition 6.14

w × I(x, y, λ) = I(wx, wy, λ)

Proof. Choose y_{0} ∈ X (δ, y). The left hand side is

(6.15) \begin{aligned}w × I(x, Ψ_{R}^{+}[y_{0}], Λ[x, y, λ, y_{0}]) & = I(x, Ψ_{R}^{+}[y_{0}], w^{−1} × Λ[x, y, λ, y_{0}]) \\ & = I(x, Ψ_{R}^{+}[y_{0}], Λ[x, y, w^{−1}λ, y_{0}]) \end{aligned}

On the right hand side we may take wy_{0} as our base point. Then we get

(6.16)  \begin{aligned}I(wx, wy, λ) & = I(wx, Ψ_{R}^{−}[wy_{0}], Λ[wx, wy, λ, wy_{0}]) \\ & = I(x, w^{−1}Ψ_{R}^{−}[wy_{0}], w^{−1}Λ[wx, wy, λ, wy_{0}]) \\ & = I(x, w^{−1}Ψ_{R}^{−}[wy_{0}], Λ[x, y, w^{−1}λ, y_{0}]) \end{aligned}

The result follows from (5.4), which says that w^{−1}Ψ_{R}^{−}[wy_{0}] = Ψ_{R}^{−}[y_{0}]. □

6.2 Parity Condition and the Cross Stabilizer

Suppose α is a real root. Then s_{α}x = x. We see that a real root α is in the cross stabilizer of π = I(x, Ψ_{R}^{−}, Λ), i.e. s_{α} × π ≃ π, if and only if

s_{α}y = y.

This is on the dual side, and says that s_{α} ∈ W (K_{y}^{∨}, H^{∨}), s_{α} is an imaginary reflection, in the cross stabilizer of the dual representation.

It is useful to compute this directly in terms of I(x, Ψ_{R}^{−}, Λ), if only as a sanity check. Recall the parity condition [8]. The roots which do not satisfy the parity condition are the + part of the cograding of real roots; I like to think of these as the “non-parity” roots, also known as the irreducibility roots. Then α fails the parity condition implies s_{α}×π = π, but not conversely.

Lemma 6.17 Fix π = I(x, Ψ_{R}^{−}, Λ) and a real root α. Write Λ = Λ(x, λ, ν) and let k = ⟨λ − ν − ρ_{R}, α^{∨}⟩.

1. s_{α} × π ≃ π if and only if sgn(α)^{k} = 1

2. α is a non-parity root if and only if k ≡ 0 (mod 2).

3. α is a non-parity root if and only if (Λ ⊗ ρ)(m_{α}) = (−1)^{⟨λ+ρ_{cx},α^{∨}⟩}.

Proof. We have

s_{α}(s_{α} × I(x, Ψ_{R}^{−}, Λ)) = I(x, s_{α}Ψ_{R}^{−}, s_{α}(s_{α} × Λ))

which by (6.3) equals

I(x, Ψ_{R}^{−}, s_{α}(s_{α} × Λ)) ⊗ τ(Ψ_{R}^{−}, α)

By (2.18) we have

(6.18) \begin{aligned}s_{α}(s_{α} × Λ(x, λ, ν)) ⊗ τ(s_{α}) & = Λ(x, λ, ν) ⊗ Λ(0, ⟨λ − ν + ρ_{R}, α^{∨}⟩α) \\ & = Λ(x, λ, ν) ⊗ sgn(a)^{k}. \end{aligned}

This proves the first claim.

If α is type II, i.e. sgn(α) = 1, then obviously s_{α} × π = π. If α is type I then this holds if and only if k ≡ 0 (mod 2). This proves (2). [Check: this is really correct.] Now

⟨λ − ν − ρ_{R}, α^{∨}⟩ ∈ 2ℤ

if and only if

⟨λ + ρ_{im} + ρ_{cx} − (ν + ρ), α^{∨}⟩ ∈ 2ℤ

Here ρ = ρ_{R} + ρ_{im} + ρ_{cx} as usual. Now Λ ⊗ ρ factors to H and it is easy to see that

(Λ ⊗ ρ)(m_{α}) = (−1)^{⟨ν+ρ,α^{∨}⟩}

On the other hand ⟨ρ_{im}, α^{∨}⟩ = 0 and therefore

(−1)^{⟨λ+ρ_{im}+ρ_{cx}−(ν+ρ),α^{∨}⟩} = (−1)^{⟨λ+ρ_{im},α^{∨}⟩}Λ(m_{α})
□

The parity condition in this result is simpler than that of [8], it doesn’t have the mysterious ϵ_{α} term. The relation involves the next result, cf. [2, Lemma 14.6]. Also see Section 7.

Lemma 6.19 Suppose Ψ^{+} satisfies: α > 0 complex implies θα < 0. Let ρ_{cx} = \frac{1}{2} ∑_{α∈Ψ_{cx}^{+}} α where Ψ_{cx}^{+} is the positive complex roots. Then ϵ_{α} = (−1)^{⟨ρ_{cx},α^{∨}⟩}.

Remark 6.20 If Ψ^{+} satisfies the conditions of the Lemma then Λ ⊗ ρ is what is denoted Γ elsewhere (cf. Section 7), but not otherwise.

6.3 Cayley Transforms

Fix x let Ψ_{R,x} be the real roots with respect to θ_{x}, and choose a set of positive real roots Ψ_{R,x}^{+}. Suppose I(x, Ψ_{R,x}^{−}, Λ) is a standard module.

Now suppose α is a non-compact imaginary root and Ψ_{R,c^{α}x}^{+} is a set of positive real roots for θ_{c^{α}x}. Then I(σ_{α}x, Ψ_{R,σ_{α}x}^{−}, c^{α}Λ) is defined.

Assume (Ψ_{R,x}^{+}, Λ) and (Ψ_{R,σ_{α}x}^{+}, c^{α}Λ) are in good position. For example take Ψ_{R,x}^{+} and Ψ_{R,σ_{α}x}^{+}. [Actually, you need to take the opposite root system… there is something to straighten out here.]

Proposition 6.21

c^{α}I(x, Ψ_{R,x}^{−}, Λ) = I(σ_{α}x, Ψ_{R,σ_{α}x}^{−}, c^{α}Λ)

Note: To be honest I’m guessing here - this could be checked from the usual definition. I’ve partly gotten it by working backwards. I’m not sure about the “good position” aspect.

Proposition 6.22 Suppose α is a noncompact imaginary root of Ψ with respect to x.

(6.23)  c^{α}I(x, y, λ) = I(σ_{α}x, σ^{α}y, λ)

Proof.

We first reduce to the case α is simple for Ψ_{im,x}^{+}. If α is not simple choose w ∈ W_{im} so that β = wα is simple.

c^{α}I(x, y, λ) = w × c^{β}I(x, y, λ)

= w × I(σ_{β}x, σ^{β}y, λ) (assuming the result for simple roots)

= I(w · σ_{β}x, w · σ^{β}y, λ) (Proposition 6.14)
= I(σ_{α}x, σ^{α}y, λ)

So assume α is simple for Ψ_{im,x}^{+}. Choose y_{0} = y_{0}[y], so

Ψ_{R}^{+}[y_{0}] = Ψ_{R,y}^{+}.

The left hand side is then

\begin{aligned}c^{α}I(x, y, λ) & = c^{α}I(x, Ψ_{R,x}^{+}, Λ[x, y, λ, y_{0}] \\ & = I(σ_{α}x, Ψ_{R,σ_{α}x}^{+}, Λ[σ_{α}x, σ^{α}y, λ, σ^{α}y_{0}]) \end{aligned}

The last equality uses the fact that (Ψ_{R,σ_{α}x}^{+}, Λ) is in good position (this only depends on λ = dΛ).

On the other hand choose y′_{0} = y_{0}[σ^{α}y], so the right hand side is

I(σ_{α}x, σ^{α}y, λ) = I(σ_{α}x, Ψ_{R,σ_{α}x}^{+}, Λ[σ_{α}x, σ^{α}y, λ, y′_{0}])

So we are done provided y′_{0} = σ^{α}y_{0}, i.e.

y_{0}[σ^{α}y] = σ^{α}y_{0}[y]

Writing this on the G side, and with w = p(y) in place of y we need

(6.24)  x_{b}[σ_{α}w] = σ_{α}x_{b}[w]

for α an imaginary noncompact root, simple for Ψ_{im,w}^{+}. This is precisely Proposition 5.18, cf. Conjecture 5.19. □

7 Relating Γ and Λ

This is taken from [7]. We relate I(Ψ_{R}^{+}, Λ) to I(Γ, λ). Here (Γ, λ) is a regular character as in [8]. We don’t need final limit parameters here.

Given I(Ψ_{R}^{+}, Λ), choose any set Ψ^{+} of positive roots satisfying Ψ_{R}^{+} ⊂ Ψ^{+} and

(7.1)  α > 0 complex ⇒ θα < 0.

Let ρ = ρ(Ψ), and let ρ_{im,c} be one half the sum of the positive imaginary compact roots. Write Λ = Λ(λ, ν). Let

Γ = Λ(λ + ρ_{i} − 2ρ_{im,c}, ν + ρ − 2ρ_{im,c})

According to [7] then have I(Λ, Ψ_{R}^{−}, λ) = I(Γ, λ).

Remark 7.2 Check this. Is the Ψ_{R}^{+} correct?

References

[1] J. Adams. Lifting of characters, volume 101 of Progress in mathematics. Birkhäuser, Boston,Basel,Berlin, 1991.

[2] J. Adams and P. Trapa. Duality for nonlinear simply laced groups.

[3] Jeffrey Adams. Parameters for representations of real groups. preprint, atlas web site.

[4] Fokko du Cloux. Combinataorics for the representation theory of reductive groups. preprint.

[5] D. Vogan J. Adams. Lifting of characters and Harish-Chandra’s method of descent. preprint.

[6] D. Vogan J. Adams. L-groups, projective representations, and the Langlands classification. 113:45–138, 1992.

[7] Annegret Paul. The dictionary of representations. preprint.

[8] D. Vogan. Irreducible characters of semisimple Lie groups IV. charactermultiplicity duality. 49, No. 4:943–1073, 1982.