Bernstein components via the Bernstein center

Abstract

Let G be a reductive p-adic group. Let $\Phi$ be an invariant distribution on G lying in the Bernstein center ${\mathcal {Z}}(G)$. We prove that $\Phi$ is supported on compact elements in G if and only if it defines a constant function on every component of the set ${\text {Irr}}(G)$; in particular, we show that the space of all elements of ${\mathcal {Z}}(G)$ supported on compact elements is a subalgebra of ${\mathcal {Z}}(G)$. Our proof is a slight modification of the argument from Section 2 of Dat (J Reine Angew Math 554:69–103, 2003), where our result is proved in one direction.

Appendix: Proof of Theorem 1.8 (by R. Bezrukavnikov)

R. Bezrukavnikov, Department of Mathematics, MIT, Cambridge, MA 02139, USA. Email: bezrukav@math.mit.edu

1.1 Decomposition of $\overline{{\mathcal {H}}}(G)$

We have an obvious perfect pairing between ${\mathcal {D}}^{\mathrm{inv}}(G)$ and $\overline{{\mathcal {H}}}(G)$. We claim that there is a decomposition

$$\begin{aligned} \overline{{\mathcal {H}}}(G)=\bigoplus \limits _{(P,\lambda )\in {\mathcal {P}}(G)} \overline{{\mathcal {H}}}(G)_{P,\lambda }, \end{aligned}$$

(4.1)

which is compatible with (1.2) by means of the above pairing. Namely, we let $\overline{{\mathcal {H}}}(G)_{P,\lambda }$ be the image of ${\mathcal {H}}(G)_{P,\lambda }$ where the latter consists of functions supported on $G_{P,\lambda }$. The fact that (4.1) holds is clear.

1.2 Spectral description of $\overline{{\mathcal {H}}}(G)$

The space $\overline{{\mathcal {H}}}(G)$ admits the following well-known description. Let $\pi \in {\mathcal {M}}(G)$ be a finitely generated representation and let E be an endomorphism of $\pi$. It is well-known (cf. e.g., [8]) that we can associate to the pair $(\pi ,E)$ and element $[\pi ,E]$ of ${\mathcal {H}}(G)$. Moreover, ${\mathcal {H}}(G)$ is isomorphic to the $\mathbb {C}$-span of symbols $[\pi ,E]$ subject to the relations:

• (a) Let $\pi _1,\pi _2\in {\mathcal {M}}(G)$ and let $u\in {\text {Hom}}(\pi _1,\pi _2), v\in {\text {Hom}}(\pi _2,\pi _1)$. Then $[\pi _1,vu]=[\pi _2,uv]$.

• (b) $[\pi _1,E_1]+[\pi _3,E_3]=[\pi _2,E_2]$ for a short exact sequence $0\rightarrow \pi _1\rightarrow \pi _2\rightarrow \pi _3\rightarrow 0$ which is compatible with the endomorphisms $E_i\in {\text {End\,}}(\pi _i)$.

• (c) $[\pi , c_1E_1+c_2E_2]=c_1[\pi ,E_1]+c_2[\pi ,E_2]$, where $c_i\in \mathbb {C}$ and $E_i\in {\text {End\,}}(\pi )$.

The action of ${\mathcal {Z}}(G)$ on $\overline{{\mathcal {H}}}(G)$ can also be described in these terms. Namely, let $\Phi \in {\mathcal {Z}}(G)$. Then $\Phi \cdot [\pi ,E]=[\pi ,E\circ \pi (\Phi )]$.

In addition, let $\rho$ be an admissible representation of G. Then we have

$$\begin{aligned} \langle [\pi ,E],{\text {ch}}_\rho \rangle =\sum \limits _i (-1)^i {\text {Tr}}(E,{\text {Ext}}^i(\pi ,\rho )). \end{aligned}$$

(4.2)

In view of the Trace Paley–Wiener theorem (cf. [2]), (4.2) defines $[\pi ,E]$ uniquely.

1.3 Spectral description of $\overline{{\mathcal {H}}}_{P,\lambda }$

Now let us fix $(P,\lambda )\in {\mathcal {P}}(G)$. Also let $\sigma$ be a finitely generated representation of M. Set

$$\begin{aligned} \pi =i_{GP}(\sigma ). \end{aligned}$$

(4.3)

Let us now choose a uniformizer t of our local field. Then any $\lambda \in Z(M)/Z(M^0)$ lifts naturally to an element $t^{\lambda }\in Z(M)$. Hence it defines an endomorphism of $\sigma$ and thus also of $\pi$. We shall denote this endomorphism by $E_{\lambda }$.

Theorem 4.4

The subspace $\overline{{\mathcal {H}}}_{P,\lambda }$ is spanned by elements $[\pi ,E_{\lambda }]$ as above.

Remark

The element $[\pi ,E_{\lambda }]$ actually depends on the choice of t; however, it is easy to see that the span of all the $[\pi ,E_{\lambda }]$ does not.

Proof

For $(P,\lambda )=(G,0)$ this is the “abstract Selberg principle” (cf. [5]). The case $P=G$ and arbitrary $\lambda$ is completely analogous.

Let us now take arbitrary P and $\lambda$. Let $\sigma$ be a finitely generated representation of the Levi group M as above and $\lambda$ – a strictly dominant cocharacter of $\pi$. Then we have a natural identification $t^{\lambda }M^0_c/ \hbox {Ad}(M)=G_{P,\lambda }/\hbox {Ad} (G)$. Hence we get a natural isomorphism between $\overline{{\mathcal {H}}}(M)_{M,0}$ and $\overline{{\mathcal {H}}}(G)_{P,\lambda }$. Indeed, if an element of $\overline{{\mathcal {H}}}(M)$ is represented by some $h\in {\mathcal {H}}(M)$ supported on $M^0_c=M_{M,0}$, then let us denote by $h_{\lambda }$ the corresponding element of ${\mathcal {H}}(M)$ supported on $M_{M,\lambda }=t^{\lambda } \cdot M_c^0$. For an open compact subgroup K of G, let us denote by $h_{\lambda ,K}$ the result of averaging $h_{\lambda }$ with respect to the adjoint action of K. Its image in $\overline{{\mathcal {H}}}(G)$ is independent of K and the assignment $h\text { mod} [{\mathcal {H}}(M),{\mathcal {H}}(M)]\mapsto h_{\lambda ,K}\text { mod} [{\mathcal {H}}(G),{\mathcal {H}}(G)]$ is the desired isomorphism. Let us denote it by $\eta _{P,\lambda }$.

Now in order to finish the proof, it is enough to show that for $\pi$ as in (4.3) we have

$$\begin{aligned}{}[\pi ,E_{\lambda }]=\eta _{P,\lambda }([\sigma ,{\text {Id}}]). \end{aligned}$$

(4.4)

Let $h=[\sigma ,{\text {Id}}]$. Then it is easy to see that

$$\begin{aligned}{}[\sigma ,t^{\lambda }]=h_{\lambda }. \end{aligned}$$

(4.5)

Now to prove (4.4) it is enough (by the Trace Paley–Wiener theorem) to check that both the LHS and the RHS of (4.4) have the same inner product with ${\text {ch}}_{\rho }$ where $\rho$ stands for an irreducible representation of G. But we have

$$\begin{aligned} {\text {Ext}}^i_G(i_{GP}(\sigma ),\rho )={\text {Ext}}^i_M(\sigma , r_{G{\overline{P}}}(\rho )). \end{aligned}$$

Hence $\langle [\pi ,E_{\lambda }],{\text {ch}}_{\rho }\rangle =\langle [\sigma ,\lambda ],r_{G{{\overline{P}}}}(\rho )\rangle$ and (4.4) follows from (4.5) and from the Casselman formula for the character of $r_{G{\overline{P}}}(\rho )$ (cf. [5]) which says for any $g\in G$ such that $P_g={{\overline{P}}}$, we have ${\text {ch}}_{\rho }(g)={\text {ch}}_{r_{G{{\overline{P}}}}(\rho )}(g)$. $\square$

Corollary 4.5

Theorem 1.8 holds.

Proof

It is enough to show that the action of any $\Phi \in {\mathcal {Z}}_{lc}(G)$ preserves every $\overline{{\mathcal {H}}}_{P,\lambda }$. Let us consider an element $[\pi ,E]$ as above; without loss of generality we may assume that all irreducible subquotients of $\sigma$ lie in one component of ${\mathcal {C}}(M)$. But then all irreducible subquotients of $\pi$ as also lie in one component $\Omega \in {\mathcal {C}}(G)$ and it follows that $\Phi \star [\pi ,E_{\lambda }]=[\pi ,f(\Phi )|_{\Omega }\cdot E_{\lambda }]=f(\Phi )|_{\Omega }\cdot [\pi ,E_{\lambda }]$ (note that $f(\Phi )|_{\Omega }\in \mathbb {C}$ as $\Phi \in {\mathcal {Z}}_{lc}(G)$). Hence the span of all the $[\pi ,E_{\lambda }]$ is preserved by the convolution with $\Phi$.