Kazhdan-Lusztig conjecture and signatures of invariant hermitian forms

David Vogan/notes by Fokko du Cloux

Notes from a lecture at MIT on December 7, 2004.

1. Setup

Let G be a connected reductive complex algebraic group, and let G be the full group of real points of some real form of G. Let {γ}_{γ∈P} be a set of Langlands parameters representing the elements of some fixed block of representations of G, with real regular infinitesimal character.

For each γ ∈ P, denote I(γ) the standard representation of G with irreducible Langlands quotient J(γ). If our representation corresponds to a Cartan H written as H = T A, where A is a vector group, T compact (usually disconnected), and write the corresponding Levi subgroup as L = MA, we may write our Langlands parameter in the form (σ, λ) where σ is a discrete series representation for M, and λ ∈ 𝔞_{R}^{∗}, 𝔞 = Lie(A). Choose a parabolic subgroup P containing L for which λ is positive. Then I(γ) may be realized as the K-finite part of Ind_{P}^{G} (σ ⊗ e^{λ} ⊗ 1).

We may embed I(γ) = I_{1}(γ) into a family I_{t}(γ), t ≥ 0, by setting I_{t}(γ) = Ind_{P}^{G} (σ ⊗ e^{tλ} ⊗ 1). Note that even when I(γ) is reducible, I_{t}(γ) will be irreducible for |t − 1| > 0 small.

2. Kazhdan-Lusztig polynomials and the Jantzen filtration

The Kazhdan-Lusztig polynomials P_{δγ}, defined for δ ≤ γ in P, where ≤ is a certain order relation on the block (or defined for any pair δ, γ, if one sets them to be 0 whenever δ and γ are not comparable), give a solution to one of the fundamental problems in representation theory, which is that of computing the multiplicity m_{δγ} of any J(δ) in a Jordan-Hölder series of I(γ) : one simply has

m_{δγ} = P_{δγ}(1)

Clearly this wastes a wealth of information contained in the P_{δγ}, and it is natural to ask for a representation-theoretic interpretation of the individual coefficients. The answer to this is given by the Jantzen filtration. This is a filtration

I(γ) = I^{0}(γ) ⊃ I^{1}(γ) ⊃ …

probably best defined in general in geometrical terms, with the property that if we denote m_{δγ}^{k} the multiplicity of J(δ) in I^{k}(γ)/I^{k+1}(γ), then one has

m_{δγ}^{k} = coefficient of q^{(l(γ)−l(δ)−k)/2} in P_{γδ}

(here l denotes a certain length function defined on the parameter set P, which is computed at the same time that one computes the order relation ≤ and the Kazhdan-Lusztig polynomials P_{δγ}.) In particular, I^{0}(γ)/I^{1}(γ) is the irreducible Langlands quotient J(γ). Note also that J(δ) can appear in I^{k} only if l(γ) − l(δ) is congruent to k modulo 2. It is known that the subquotients of the Jantzen filtration are always completely reducible.

It is well-known that J(γ) has an invariant hermitian form if and only if there is an element in the Weyl group W (G, H) fixing σ and taking λ to −λ; assume in the sequel that this symmetry condition is fulfilled. The invariant hermitian form is then unique up to a non-zero real scalar; in particular its signature is well-defined up to a global sign, and it is well-defined if we ask that the signature on a particular lowest K-type is > 0 (when there are several lowest K-types, we have to make an arbitrary choice of one of them.)

One may realize all the I_{t}(γ) as the same K-module V, with an action of 𝔤 which has an affine dependence on t. Then the hermitian forms ⟨ , ⟩_{t} which are well-defined for the irreducible I_{t}(γ) also have a nice dependency on t, and extend to degenerate) hermitian forms at the points of reducibility. It turns out that the kernel of the form ⟨ , ⟩_{1} is exactly I^{1}(γ); so we get a non-degenerate form on J(γ) by passing to the quotient.

More generally, it may be shown that I^{k} is the set of values at t = 1 of V -valued polynomial functions v(t), with the property that vor each fixed w ∈ V, the function t → ⟨v(t), w⟩_{t} vanishes at order at least k at t = 1. Then the number

lim_{t→1} \frac{⟨v(t), w⟩_{t}}{(1 − t)^{k}}

depends only on v(0), and is zero when v(0) is in I^{k+1}; in this way one may define a non-degenerate invariant hermitian form on I^{k}/I^{k+1}, which we denote ⟨ , ⟩_{1}^{k}

It should be clear that the signature of the form is lower continuous in t (when t approaches 1 from below, we are dividing by something > 0); hence we have

sign⟨ , ⟩_{t} = ∑_{k=0}^{∞} sign⟨ , ⟩_{1}^{k}  (1)

for all t < 1 close to 1. On the other hand, only the odd powers of (1 − t) change sign when we cross over to the other side, so

sign⟨ , ⟩_{t} = ∑_{k=0}^{∞}(−1)^{k}sign⟨ , ⟩_{1}^{k}  (2)

for t > 1 close to 1.

Now assume for instance that we know that I_{0}(γ) is irreducible. Then certainly the form ⟨ , ⟩_{t} is positive definite up to the first point of reducibility t_{0}, and at that point, all the ⟨ , ⟩_{t_{0}}^{k} are also positive definite (and hence all the subquotients are unitary.) From formula (2), we now know the signature of ⟨ , ⟩_{t} up to the next point of reducibility t_{1}. The general problem is that of crossing a point of reducibility; we might as well consider that at t = 1.

So look at formula (1). It expresses the left hand side, which is known, as a sum of signatures. We have essentially no information on the ⟨ , ⟩_{1}^{0} term; in fact, one of our goals is to compute it. On the other hand, we may assume that we know the signature of the normalized invariant hermitian forms on the other terms in the sum, arguing by induction (they correspond to representations that are encountered “earlier”, say in terms of the length of the parameter, where length 0 corresponds to discrete series representations). We may also assume that we know the Kazhdan-Lusztig polynomials. This allows us then to express each sign⟨ , ⟩_{1}^{k}, k > 0, in the form

sign⟨ , ⟩_{1}^{k} = ∑_{j} ε_{jk}S_{j}  (3)

where the ε_{jk} are signs, and the S_{j} are known normalized signatures of irreducible representations.

The question is: are we capable of determining these signs? The point is that most of these sign choices give rise to signature functions for J(γ) that take on negative values for certain K-types μ, and hence are impossible. Here is a very simple example. Assume that μ occurs in J(γ), and in one single J(δ) at the first level; assume moreover that the multiplicity of μ in J(γ) is 1. Then if the signature of the form on ⟨ , ⟩_{t}, t < 1 for μ is (p, q), the signature for J(δ) is either (p − 1, q) or (p, q − 1). The only case where this does not determine the sign for J(δ) is when these two candidate signatures are of opposite sign, i.e., when p = q.

Example: SL_{2}(R). This example is way too easy, but it is fun. Look at the spherical principal series representations for G = SL_{2}(R), with real infinitesimal character. They are parametrized by t ≥ 0, with an irreducible unitary principal series representation for t = 0, and points of reducibility at odd integral values of t. The signature function on the principal series representation is the datum of a sign τ_{n} for each n ∈ 2Z; the normalized signature functions have τ_{0} = +. At t = 2m+1, we have a finite-dimensional quotient F_{2m} of dimension 2m + 1. The Jantzen filtration has only two levels; we have I_{2m+1}^{1} = δ_{2m+1} ⊕ δ_{−2m−1}, where for each non-zero integer n we denote δ_{n} the discrete series with parameter n. The normalized signature function for δ_{n} has support the set of integers m with |m| > |n|, m ≡ n + 1 mod 2, and sgn(m) = sgn(n), and all plus signs of course.

Now from formula (1), at the first reducibility point we get that the trivial representation is unitary, and from formula (2), that for 1 < t < 3 the signature of ⟨ , ⟩_{t} is + for n = 0, − for n ≠ 0, n even. When we get to τ = 3, we see that for |n| > 1, the K-type 2n does not even occur in F_{2}; hence the signature on the δ_{±3} factor of I_{3}^{1} has to be all minus signs. So the signature on F_{3} is (−, +, −) for n = −2, 0, 2, and for I_{t}(γ), 3 < t < 5 it is (−, +, −) for n = −2, 0, 2, and plus for all other even values of n. It is clear how to continue.

For the non-spherical principal series, the difficulty is how to start, as the value at 0 is not irreducible. The reducibility points here are at even integral values of t. In fact, it is clear that ⟨ , ⟩_{t} cannot be positive definite for 0 < t < 2, because otherwise letting t → 2 the finite-dimensional representation F_{1} would be unitary, which it is not. So the signature on I_{0}(γ) is the one where one of the two irreducible factors is positive definite, the other negative definite (it is an arbitrary choice to decide which; say τ_{1} = +.) Now all goes on as before.

3. More about the nature of the signature functions

We have not been very precise about what the signature function actually looks like. For every admissible (𝔤, K)-module V with an invariant hermitian form, we define its signature to be a triple (p, q, z) of functions on K^{∧}, with nonnegative integer values; here p(μ), q(μ) and z(mu) will denote the signature (number of plus signs, minus signs, and dimension of the kernel) of the hermitian form induced on the multiplicity space Hom_{K}(μ, V). Of course for a non-degenerate form the function z is identically zero.

Let us further denote by χ_{K}(V) the character of V as a K-module; this is just the function which to each μ assigns the multiplicity of μ in V. Clearly we have p + q + z = χ_{K}(V). The K-character χ_{K}(I(σ, λ)) clearly depends only on σ; moreover, when λ = 0, the unitary representation I(σ) = I(σ, 0) breaks up as a direct sum J_{1} ⊕…⊕J_{r} of unitary irreducible representations, with one term for each lowest K-type appearing in I(σ).

Theorem. — For each γ, the functions p and q for I(γ) and J(γ) are Z-linear combinations of the K-characters of the irreducible components of the various I(σ) (possibly coming from other conjugacy classes of Cartan subgroups of G), such that the infinitesimal character of I(σ) is contained in the convex hull of the W -orbit corresponding to the infinitesimal character of I(γ). Idea of proof: First note that for each irreducible J(γ), χ_{K}(J(γ)) is a Zlinear combination of the χ_{K}(I(γ)) (and even that this expression can be explicitly computed if we know the Kazhdan-Lusztig polynomials.) Now note that the result is true for I_{0}(γ), by definition. Then, starting from an arbitrary I(γ), move from I_{0}(γ) to I(γ) = I_{1}(γ); each time one crosses a reduction point, the signature of J(γ) gets expressed as a difference of expressions that are known to be of the required form, and the signature for I_{t}(γ) beyond the reduction point is then also of the required form. The issue that is perhaps not entirely clear is the behaviour near 0; as shown already by the example of SL(2, R), the form ⟨ , ⟩_{t} may well be indefinite near 0, and it should be shown that necessarily the signature is given by some sign on each lowest K-type (well-defined up to a global sign change). Also, in order for the algorithm to start, we should be able to determine this initial configuration computationally.