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LSpice
LSpice
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For$\DeclareMathOperator\Res{Res}$For classical groups, V. Heiermann has proved that $Res^G_{G^0}(\pi)$$\Res^G_{G^0}(\pi)$ has no multiplicity, see :

In general, the endomorphism algebra $R$ is a free module of rank $m^2$ over its center Z$Z$, where $m$ is the multiplicity of an irreducible in $Res^G_{G^0}(\pi)$$\Res^G_{G^0}(\pi)$. See p. 181 in my book "Représentations des groupes réductifs $p$-adiques", SMF, cours spécialisés 17. I suspect that $R$ and $Z$ could be Morita equivalent.

$Z$ is always indeed isomorphic to the ring of algebraic functions on a product of GL(1,C)$\operatorname{GL}(1,C)$'s.

For classical groups, V. Heiermann has proved that $Res^G_{G^0}(\pi)$ has no multiplicity, see :

  • Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif p-adique - le cas des groupes classiques - Selecta Mathematica.

In general, the endomorphism algebra $R$ is a free module of rank $m^2$ over its center Z, where $m$ is the multiplicity of an irreducible in $Res^G_{G^0}(\pi)$. See p. 181 in my book "Représentations des groupes réductifs $p$-adiques", SMF, cours spécialisés 17. I suspect that $R$ and $Z$ could be Morita equivalent.

$Z$ is always indeed isomorphic to the ring of algebraic functions on a product of GL(1,C)'s.

$\DeclareMathOperator\Res{Res}$For classical groups, V. Heiermann has proved that $\Res^G_{G^0}(\pi)$ has no multiplicity, see :

In general, the endomorphism algebra $R$ is a free module of rank $m^2$ over its center $Z$, where $m$ is the multiplicity of an irreducible in $\Res^G_{G^0}(\pi)$. See p. 181 in my book "Représentations des groupes réductifs $p$-adiques", SMF, cours spécialisés 17. I suspect that $R$ and $Z$ could be Morita equivalent.

$Z$ is always indeed isomorphic to the ring of algebraic functions on a product of $\operatorname{GL}(1,C)$'s.

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David Renard
David Renard
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For classical groups, V. Heiermann has proved that $Res^G_{G^0}(\pi)$ has no multiplicity, see :

  • Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif p-adique - le cas des groupes classiques - Selecta Mathematica.

In general, the endomorphism algebra $R$ is a free module of rank $m^2$ over its center Z, where $m$ is the multiplicity of an irreducible in $Res^G_{G^0}(\pi)$. See p. 181 in my book "Représentations des groupes réductifs $p$-adiques", SMF, cours spécialisés 17. I suspect that $R$ and $Z$ could be Morita equivalent.

$Z$ is always indeed isomorphic to the ring of algebraic functions on a product of GL(1,C)'s.