Parameters for twisted representations

Abstract

The main result of [4] is the description of an algorithm to compute the signature of the Hermitian form on an irreducible representation of a real reductive Lie group G, and therefore determine if it is unitary. This paper concerns an important ingredient of the algorithm. If the inner class of G is defined by an outer automorphism δ, so that G does not have discrete series representations, it is necessary to compute a new class of Kazhdan–Lusztig–Vogan polynomials for G. These were defined and studied by Lusztig and Vogan in [10]. In order to carry out the computation, we introduce new class of twisted parameters, and study the Hecke algebra action in the resulting basis.

Appendix

We collect a few technical results about the Tits group [13], which will be needed for our study of parameters for representations in Section 3. We continue with the notation of (11). For each simple root α, the pinning defines a canonical homomorphism

$$\displaystyle{ \phi _{\alpha }: (\mathrm{SL}(2),\mbox{ diag}) \rightarrow (G,H)\qquad d\phi _{\alpha }\left (\begin{array}{*{10}c} 0&1\\ 0 &0 \end{array} \right ) = X_{\alpha }. }$$

(53a)

Similarly,

$$\displaystyle{ \phi _{\alpha ^{\vee }}: (\mathrm{SL}(2),\mbox{ diag}) \rightarrow (^{\vee }G,^{\vee }H)\qquad d\phi _{\alpha ^{\vee }}\left (\begin{array}{*{10}c} 0&1\\ 0 &0 \end{array} \right ) = X_{\alpha ^{\vee }}. }$$

(53b)

It is sometimes convenient to define also

$$\displaystyle{ H_{\alpha } = d\phi _{\alpha }\left (\begin{array}{*{10}c} 1& 0\\ 0 &-1 \end{array} \right ),\qquad X_{-\alpha } = d\phi _{\alpha }\left (\begin{array}{*{10}c} 0&0\\ 1 &0 \end{array} \right ); }$$

(53c)

the first element (because α(H _α ) = 2) “is” the coroot α ^∨. The second is a preferred root vector for −α, characterized by the last of the three relations

$$\displaystyle{ \left [H_{\alpha },X_{\alpha }\right ] = 2X_{\alpha },\quad \left [H_{\alpha },X_{-\alpha }\right ] = -2X_{-\alpha },\quad \left [X_{\alpha },X_{-\alpha }\right ] = H_{\alpha }. }$$

(53d)

In this way we get a distinguished representative

$$\displaystyle{ \begin{array}{ll} \sigma _{\alpha } & = _{\mbox{ def}}\phi _{\alpha }\left (\begin{array}{*{10}c} 0 &1\\ -1 &0 \end{array} \right ) =\exp ( \frac{\pi }{2}(X_{\alpha } - X_{-\alpha })) \\ \qquad \sigma _{\alpha }^{2} & = m_{\alpha } = _{\mbox{ def}}\alpha ^{\vee }(-1)\end{array} }$$

(53e)

for the simple reflection s _α . These representatives satisfy the braid relations (see [13]) and therefore define distinguished representatives

$$\displaystyle{ \sigma _{w} = _{\mbox{ def}}\sigma _{\alpha _{1}}\sigma _{\alpha _{2}}\cdots \sigma _{\alpha _{r}}\qquad (w = s_{\alpha _{1}}s_{\alpha _{2}}\cdots s_{\alpha _{r}}\ \mbox{ reduced}) }$$

(53f)

for each Weyl group element w. (That $\sigma _{w}$ is independent of the choice of reduced decomposition is a consequence of the fact that the $\sigma _{\alpha }$ satisfy the braid relations.) If γ is any distinguished (that is, pinning-preserving) automorphism of (G, B, H), then

$$\displaystyle{ \gamma (\sigma _{w}) =\sigma _{\gamma (w)}. }$$

(53g)

The braid relations imply, for any w ∈ W and simple root α

$$\displaystyle{ \sigma _{w}\sigma _{\alpha } = \left \{\begin{array}{@{}l@{\quad }l@{}} \sigma _{ws_{\alpha }} \quad &\mbox{ length}(ws_{\alpha }) = \mbox{ length}(w) + 1 \\ \sigma _{ws_{\alpha }}m_{\alpha }\quad &\mbox{ length}(ws_{\alpha }) = \mbox{ length}(w) - 1 \end{array} \right. }$$

(53h)

and a similar result for $\sigma _{\alpha }\sigma _{w}$ (with m _α on the left).

In exactly the same way, we get a distinguished representative in^∨ G:

$$\displaystyle{ ^{\vee }\sigma _{ w} = _{\mbox{ def}}\sigma _{\alpha _{1}^{\vee }}\sigma _{\alpha _{2}^{\vee }}\cdots \sigma _{\alpha _{r}^{\vee }}\qquad (w = s_{\alpha _{1}}s_{\alpha _{2}}\cdots s_{\alpha _{r}}\ \mbox{ reduced}). }$$

(53i)

The main fact we need about these representatives is

Proposition 12.1.

In the setting of (53) ,

$$\displaystyle{\begin{array}{ll} \sigma _{w}\sigma _{w^{-1}} & = (w\rho ^{\vee }-\rho ^{\vee })(-1) \\ & = e((\rho ^{\vee }- w\rho ^{\vee })/2)\\ &=\prod \limits _{ \begin{array}{c}\beta \in R^{+}(G,H) \\ w^{-1}\beta \notin R^{+}(G,H)\end{array}}m_{\beta }. \end{array} }$$

The proof is an easy induction on ℓ(w). See [2, Lemma 5.4].

Proposition 12.2.

In the setting (11) , suppose w ∈ W, α,β ∈Π are simple roots, and wα = β. Write X _α and X _β for the simple root vectors given by the pinning, and $\sigma _{w} \in N(H)$ for the Tits representative of w defined in (53f) . Then

$$\displaystyle{\sigma _{w}\sigma _{\alpha }\sigma _{w}^{-1} =\sigma _{\beta }}$$

and

$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{\alpha }) = X_{\beta },\quad \mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{-\alpha }) = X_{-\beta }.& &{}\end{array}$$

(54)

Proof.

Since β = w α, $s_{\beta }w = ws_{\alpha }$. If $\mbox{ length}(ws_{\alpha }) = \mbox{ length}(s_{\beta }w) = \mbox{ length}(w) + 1$, then the first case of (53h) implies

$$\displaystyle{\sigma _{w}\sigma _{\alpha } =\sigma _{ws_{\alpha }} =\sigma _{s_{\beta }w} =\sigma _{\beta }\sigma _{w}.}$$

If the lengths are decreasing, we see

$$\displaystyle{\begin{array}{ll} \sigma _{w}\sigma _{\alpha }& =\sigma _{ws_{\alpha }}m_{\alpha }\\ & =\sigma _{ s_{\beta }w}m_{\alpha }\\ & = m_{\beta }\sigma _{\beta }\sigma _{ w}m_{\alpha }\\ & = m_{\beta }m_{ s_{\beta }w\alpha }\sigma _{\beta }\sigma _{w} \\ & =\sigma _{\beta }\sigma _{w}\quad \mbox{ (since $s_{\beta }w\alpha = -\beta $)}. \end{array} }$$

For the second statement we observe that $\mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{\alpha })$ is some multiple of X _β . The Tits group preserves the $\mathbb{Z}$-form of $\mathfrak{g}$ generated by the various X _±α , so this scalar is ± 1; we need to show it is 1. We compute

$$\displaystyle{\begin{array}{ll} \sigma _{w}\sigma _{\alpha }\sigma _{w}^{-1} & =\sigma _{w}(\exp \frac{\pi }{2}(X_{\alpha } - X_{-\alpha }))\sigma _{w}^{-1} \\ & =\exp ( \frac{\pi }{2}\mathop{ \mathrm{Ad}}\nolimits (\sigma _{w})(X_{\alpha } - X_{-\alpha })). \end{array} }$$

On the other hand, by what we just proved this equals

$$\displaystyle{\sigma _{\beta } =\exp ( \frac{\pi } {2}(X_{\beta } - X_{-\beta })).}$$

Setting these equal gives the two equalities in the second statement. □ 

Corollary 12.3.

In the setting (11), suppose w ∈ W, α,β ∈Π are simple roots, and wα = −β. Write X _α and X _β for the simple root vectors given by the pinning, and $\sigma _{w} \in N(H)$ for the Tits representative of w defined in (53f) . Then

$$\displaystyle{\sigma _{w}\sigma _{\alpha }\sigma _{w}^{-1} =\sigma _{\beta }}$$

and

$$\displaystyle{\mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{\alpha }) = -X_{-\beta },\qquad \mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{-\alpha }) = -X_{\beta }.}$$

Proof.

Let w′ = ws _α . The first assertion follows from the previous lemma applied to $\sigma _{w'}$, using the fact that $\sigma _{w'} =\sigma _{w}\sigma _{\alpha }$ (since w′ = ws _α is a reduced expression). As in the proof of the previous proposition we conclude that

$$\displaystyle{\exp ( \frac{\pi } {2}(\mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{\alpha } - X_{-\alpha })) =\exp ( \frac{\pi } {2}(X_{\beta } - X_{-\beta })),}$$

and in this case this implies $\mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{\alpha }) = -X_{-\beta }$ and $\mathop{\mathrm{Ad}}\nolimits (\sigma _{w})(X_{-\alpha }) = -X_{\beta }$. □