Abstract
Let G be a reductive padic 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.
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References
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Dedicated to J. Bernstein on the occasion of his 70th birthday
R.B. was partially supported by the National Science Foundation Grant DMS1102434. D.K. was partially supported by the European Research Council. A.B, D.K., R.B all supported by the USIsrael Binational Science Foundation. Appendix section contributed by R. Bezrukavnikov.
Appendix: Proof of Theorem 1.8 (by R. Bezrukavnikov)
R. Bezrukavnikov, Department of Mathematics, MIT, Cambridge, MA 02139, USA. Email: bezrukav@math.mit.edu
Appendix: Proof of Theorem 1.8 (by R. Bezrukavnikov)
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
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 wellknown description. Let \(\pi \in {\mathcal {M}}(G)\) be a finitely generated representation and let E be an endomorphism of \(\pi \). It is wellknown (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
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
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
Let \(h=[\sigma ,{\text {Id}}]\). Then it is easy to see that
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
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 \(\Phi \).