Neighborhoods at infinity and the Plancherel formula for a reductive p-adic symmetric space

Abstract

Yiannis Sakellaridis and Akshay Venkatesh have determined, when the group G is split and the field $\mathbf {F}$ is of characteristic zero, the Plancherel formula for any spherical space X for G modulo the knowledge of the discrete spectrum. The starting point is the determination of good neighborhoods at infinity of X / J, where J is a small compact open subgroup of G. These neighborhoods are related to “boundary degenerations” of X. The proof of their existence is made by using wonderful compactifications. In this article we show the existence of such neighborhoods assuming that $\mathbf {F}$ is of characteristic different from 2 and X is symmetric. In particular, one does not assume that G is split. Our main tools are the Cartan decomposition of Benoist and Oh, our previous definition of the constant term and asymptotic properties of Eisenstein integrals due to Nathalie Lagier. Once the existence of these neighborhoods at infinity of X is established, the analog of the work of Sakellaridis and Venkatesh is straightforward and leads to the Plancherel formula for X.

Appendix: Rational representations

In this section we establish some results on rational representations of G which are needed to extend the results of [12] and [4], which are established when $\mathbf {F}$ is of characteristic zero, to the case where $\mathbf {F}$ is simply of characteristic different from 2.

1.1 Rational representations and parabolic subgroups

Let $\underline{G}$ be a reductive algebraic group defined over a non archimedean local field $\mathbf {F}$, whose group of $\mathbf {F}$-points is equal to G. We will use similar notations for the subgroups of G.

Let $A_0$ be a maximal split torus of G and let $P_0=M_0U_0$ be a minimal parabolic subgroup of G with $A_0\subset M_0$. Let T be a maximal $\mathbf {F}$-torus of $\underline{G}$ which contains $\underline{A}_0$. Let B be a Borel subgroup of $\underline{G}$, which contains T and is contained in $\underline{P}_0$. One denotes by $\Sigma (T)$ the set of roots of T in the Lie algebra of $\underline{G}$. One denotes by $\Lambda (T)$ (resp., $\Lambda (T)_{rac}$) the weight lattice (resp., the root lattice) of T with respect to $\underline{G}$. We adopt similar notations for $A_0$. Let $\Gamma$ be the absolute Galois group of $\mathbf {F}$ which acts on these lattices. Let $\Lambda ^+(T)$ be the set of dominant weights for T relative to B. Let $\Lambda ^+(A_0)$ (resp., $\Lambda ^+(A_0)_{rac}$) the set of dominant elements for $P_0$ of $\Lambda (A_0)$ (resp., $\Lambda (A_0)_{rac}$).

Definition 6

Let $P=MU$ be a parabolic subgroup of G which contains $P_0$, where M is its Levi subgroup which contains $A_0$. One denotes by $\Lambda ^+_M(T)$ the set of elements $\lambda$ of $\Lambda ^+(T)$such that G has a rational finite dimensional irreducible representation, defined over $\mathbf {F}$, with highest weight $\lambda$ relative to $B, (\pi _\lambda , V_\lambda )$, with the following property:

The goal of this subsection is to produce sufficiently many elements of $\Lambda ^+_M(T)$.

Proposition 9

• (i) Let $T_{an}$ be the anisotropic component of T. There exists $n\in \mathbb {N}^*$ such that any element $\lambda$ of $n\Lambda ^+(A_0)$ extends uniquely to an element $\mu$ of $\Lambda (T)_{rac}$ trivial on $T_{an}$.

• (ii) If $\lambda$ is orthogonal to the simple roots of $A_0$ in the Lie algebra of $U_0\cap M$ then $\mu$ is element of $\Lambda ^+_M(T)$.

For the proof we need several lemmas.

Let $\beta$ be an element of the set, $\Sigma (A_0)$, of roots of $A_0$ in the Lie algebra of G. One defines:

$$\begin{aligned} \underline{\beta }: = \sum _{ \alpha \in \Sigma (T), \alpha _{\vert A_0}= \beta } \alpha . \end{aligned}$$

One sees easily that:

We fix, once for all, such integers $n'$ and $n'_\beta$

Lemma 18

Every element $\lambda$ of $n'\Lambda _{rac}(A_0)$ extends uniquely to an element $\mu$ of $\Lambda _{rac}(T)$ trivial on the anisotropic component $T_{an}$ of T, invariant by $\Gamma$ and by $W(\underline{M}_0, T)$.

Let us denote by $(n'\Lambda _{rac}(A_0))^{\tilde{}}$ the lattice generated by the $n'_\beta \underline{\beta }, \beta \in \Sigma (A_0)$. From their definition, one sees that the elements of $(n'\Lambda _{rac}(A_0))^{\tilde{}}$ are invariant under $\Gamma$ and are elements of $\Lambda _{rac}(T)$. One remarks that every element $\mu$ of $(n'\Lambda _{rac}(A_0))^{\tilde{}}$ is invariant by the Weyl group of $\underline{M}_0$ relative to $T, W(\underline{M}_0 , T)$.

Let us show any element $\mu$ is trivial on $T_{an}$. One can choose T such that it contains a maximal torus defined over $\mathbf {F}, T_1$, of the derived group of $\underline{M}_0$. Actually, by conjugacy, one sees that any T has this property. Moreover T contains the maximal anisotropic torus $C_{an}$ of the center of $M_0$. The product $T_1 C_{an} \underline{A}_0$ is a torus. For reasons of dimension it is a maximal torus of G. Hence $T= T_1 C_{an} \underline{A}_0$. Notice that $T_1 C_{an}$ is the anisotropic component $T_{an}$ of T. As $\mu$ is $W(\underline{M}_0 , T)$-invariant, the restriction of $\mu$ to $T_1$ is trivial. As $C_{an}$ is anisotropic, the invariance by $\Gamma$ of $\mu$ shows that its restriction to $C_{an}$ is trivial. This proves the existence part of the Lemma. As $T= T_{an} \underline{A}_0$ the unicity follows. $\square$

Lemma 19

• (i) There exists $n\in \mathbb {N}$ such that $n\Lambda (A_0) \subset n' \Lambda _{rac}(A_0)$.

• (ii) If $\lambda \in n\Lambda ^+(A_0)$, its extension $\mu$ to T given by the preceding lemma is the highest weight of a rational representation of G, defined over $\mathbf {F}$, denoted $(\pi _{\mu }, V_{\mu } ).$

Proof

• (i) The lattice $n'\Lambda _{rac}(A_0)$ is contained in the lattice $\Lambda (A_0)$. As these lattices are of the same rank, there exists $n\in \mathbb {N}^*$ such that $n\Lambda (A_0) \subset n'\Lambda _{rac}(A_0).$

• (ii) From (i) and the preceding lemma, if $\lambda \in n\Lambda ^+(A_0), \mu$ is in $\Lambda _{rac}(T) \subset \Lambda (T)$. Moreover if $\alpha$ is a root of T in the Lie algebra of $\underline{G}, \langle \mu ,\AA \rangle =\langle \lambda ,\alpha _{\vert A_0} \rangle$. Hence $\mu$ is a dominant weight. From the preceding Lemma, it is invariant by $\Gamma$. Then [15], Theorem 3.3 and Lemma 3.2 implies (ii). $\square$

Lemma 20

Let $\lambda \in n \Lambda ^+(A_0)$ and $\mu$ as in Lemma 18. Then, with the notation of the preceding lemma, $M_0$ acts on a non zero highest weight vector of $(\pi _\mu , V_\mu )$ by a rational character of $M_0$ again denoted by $\mu$.

Proof

As $\pi _\mu$ is defined over $\mathbf {F}$, it is enough to prove that $v_\mu$ transforms under a rational character of $M_0$. In order to prove this, one can work with the algebraic closure. The invariance of $\mu$ by $W(\underline{M}_0 , T)$ (cf. Lemma 18), the fact that the space of weight $\mu$ in $V_\mu$ is of dimension one (cf. [9], Proposition 33.2) together with the Bruhat decomposition of $\underline{M}_0$ allow to prove the Lemma. $\square$

Proof of Proposition 9

(i) follows from Lemmas 18 and  19.

Let $\lambda \in n \Lambda ^+{(A_0)}$ be as in the statement of Proposition 9(ii) i.e. $\lambda$ is orthogonal to the simple roots of $A_0$ in the Lie algebra of $U_0^-\cap M$. Let $\mu$ be as in Lemma 18. Let $(\pi _\mu , V_\mu )$ be as in Lemma 19, and let $v_\mu$ be a non zero highest weight vector. One has to prove that $v_\mu$ transforms under M by a rational character of M that we will still denote by $\mu$. It is enough to prove that the line $\mathbf {F}\mu$ is stable by the action of M. One shows, using the preceding Lemma, by a proof analogous to the one of [9], Proposition 31.2 and using the density of $U_0^-M_0U_0$ in G, that the $A_0$-weight space of $V_\mu$ for the weight $\lambda$ is one dimensional. The Weyl group, $W(M, A_0)$, of M relative to $A_0$ fixes $\lambda$ from the hypothesis on $\lambda$. One finishes the proof of our assertion on the action M on $v_\mu$ by using the Bruhat decomposition of M relative to $P_0\cap M$. Hence $\mu \in \Lambda ^+_M(T)$.

$\square$

1.2 H-distinguished rational representations of G

Proposition 9 allows to extend the results of [4] section 2.7 and especially Propositions 2.9, 2.11 to a non archimedean local field, $\mathbf {F}$, of characteristic different from 2. Let $\Sigma (G, A_{0})$ (resp., $\Sigma (P_{0}, A_{0})$ or $\Sigma (P_{0})$) the set of roots of $A_0$ in the Lie algebra of G (resp., $P_0$). We denote by $\Delta (P_{0})$ the set of simple roots of $\Sigma (P_{0})$.

Let $P=MU$ be a standard $\sigma$-parabolic subgroup of G. We will use the notation of the main body of the article. Let us assume that $A_\emptyset \subset A_0$, which is automatically $\sigma$-stable, and $P_0\subset P_\emptyset$. Let $\{ \alpha _1, \dots , \alpha _{m_0} \}$ be the simple roots of $\Sigma (P_{0})$ written in such a way that $\{ \alpha _1, \dots , \alpha _{m_\emptyset } \}$ are the simple roots in the Lie algebra of $U_\emptyset , \{ \alpha _1, \dots , \alpha _{m} \}$ are the simple roots in the Lie algebra of U. One has the fundamental weights of $\Sigma (P_{0}, A_{0}), \delta _1, \dots , \delta _l$.

Let $i=1,\dots , m$ and $\lambda _i= n\delta _i$ with n as in Proposition 9. From this proposition, $\lambda _i \in \Lambda ^+_{M}(T)$ and there exists a unique rational character of $T, \mu$, trivial on $T_{an}$ and whose restriction to $A_0$ is equal to $\lambda _i$ and $\mu$ is the highest weight of an irreducible finite dimensional rational representation of G, denoted by $(\pi _\mu , V_\mu )$. Moreover if $v_\mu$ is a non zero highest weight vector in $V_\mu$, the space $\mathbf {F}v_\mu$ is P-invariant. We denote again by $\mu$ the rational character of M which describes the action of M on $v_\mu$. One denotes by $v'_\mu$ the unique element of $V_{\mu }'$ of weight $\mu ^{-1}$ under M and such that $\langle v'_\mu ,v_\mu \rangle = 1$.

Let $\nu :=\mu (\mu ^{-1}\circ \sigma )\in \Lambda (T)$ and let $(\tilde{\pi }_\nu , \tilde{V}_\nu )$ be the rational representation of G $(\pi _\mu \otimes (\pi _\mu '\circ \sigma ),V_{\mu }\otimes V_{\mu }')$. Let $\tilde{v}_\nu :=v_\mu \otimes v_\mu '$ which is of weight $\nu$ under the representation $\tilde{\pi }_\nu$ restricted to M. Then there exists a non zero H-invariant vector, under $\tilde{\pi }_\nu$ in $\tilde{V}_\nu '=(V_{\mu }\otimes V_{\mu } ')'\simeq V_{\mu }'\otimes V_{\mu } \simeq End (V_\mu )$, namely the identity that we will denote $e_{\nu , H}'$. It satisfies $\langle e_{\nu ,H}', \tilde{v}_\nu \rangle = 1$.

Let us show that $\nu = 2 \mu$. As $\sigma$ preserves $T_{an}$, the character $\mu ^{-1}\circ \sigma$ of T is trivial on $T_{an}$. Its restriction to $A_0$ is equal to $\lambda$. From the unicity statement of $\mu$, it is equal to $\mu$. This proves our claim.

From this it follows that