5
$\begingroup$

$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$.

Recall the following standard result [1, Thm. 10.23]:

Theorem. Let $L_i$ be an $RH_i$-module, $i=1,2$. Then $$\Hom_{RG}(L_1^G,L_2^G) \simeq \bigoplus_{x^{-1}y \in D} \Hom_{R({^xH_1} \cap {^yH_2})}({^xL_1}, {^yL_2})$$ as $R$-modules, where the sum is taken over all $D \in H_1 \backslash G/H_2$.

Is there an analogue of the above theorem for ordinary smooth representations of a locally profinite group (l.p.g. for short)?

I have already checked introductory chapters of some classical references in ordinary smooth representation theory of l.p.g.'s [2, 3, 4] but without success.

The closest hint I was given seems to be the general discussion in [5, Sec. 5.5], but unfortunately I cannot access it freely on the Internet or in any library from my university. Thank you in advance.

Bibliography

[1] C. W. CURTIS AND I. REINER, Representation Theory of Finite Groups and Associative Algebras, John Wiley and Sons Inc, 1962.

[2] C. J. BUSHNELL AND G. HENNIART, The Local Langlands Conjecture for GL(2), Springer, 2006.

[3] W. CASSELMAN, Introduction to the theory of admissible representations of p-adic reductive groups (1974), Preprint, University of British Columbia.

[4] I.N. BERNSTEIN AND A.V. ZELEVINSKY, Representations of the group GL(n, F) where F is a local non-archimedean field, Uspekhi Mat. Nauk 31 (1976), 5–70.

[5] M.-F. VIGNÉRAS, Représentations l-modulaires d'un groupe réductif p-adique avec l ≠ p, Progress in Math. 137, Birkhaüser, 1996.

$\endgroup$
4
  • $\begingroup$ You'll probably have a lot better luck if you state, or at least summarise, the theorem that you want generalised. $\endgroup$
    – LSpice
    Oct 11, 2019 at 17:41
  • $\begingroup$ Crosspost of math.stackexchange.com/questions/3388557 $\endgroup$
    – YCor
    Oct 12, 2019 at 9:28
  • $\begingroup$ Your notation $\bigoplus_{x^{-1}y \in D}$ (and the fact that the summands depend on $x$ and $y$) suggests that you are summing over pairs $(x, y)$, but then you say that you are summing over $D$. Which one is it—or should it be a double sum, over $D$ and then over $(x, y)$? $\endgroup$
    – LSpice
    Oct 12, 2019 at 21:54
  • $\begingroup$ @LSpice I apologize for the abuse of notation. For each pair $(x,y)$ of elements in $G$, it can be shown that the $R-$module $Hom_{R(^xH_1 \cap ^yH_2)}(^xL_1, ^yL_2)$ depends only on the double coset $D \in H_1 \backslash G / H_2$ to which $x^{-1}y$ belongs. The sum is taken over all double cosets $D$, where for each $D$ we choose a pair $(x,y)$ of elements in $G$ such that $x^{-1}y \in D$. $\endgroup$ Oct 13, 2019 at 0:25

2 Answers 2

4
$\begingroup$

Let me reproduce and translate the relevant results from Vignéras's book (keeping the same notations). Putting these results together yields some generalisation of what you stated, but you have to be careful with the two types of induction. $\newcommand{\Mod}{\mathrm{Mod}} \newcommand{\Res}{\mathrm{Res}} \newcommand{\Ind}{\mathrm{Ind}} \newcommand{\ind}{\mathrm{ind}}$

Let $R$ be a commutative unital ring. Let $G$ be a locally profinite group. Let $\Mod_R(G)$ denote the category of smooth $RG$-modules.

Section 5.1:

Let $H$ be a closed subgroup of $G$. Let $$ \Res_{H,G}\colon \Mod_R(G) \to \Mod_R(H) $$ denote the restriction functor.

Let $$ \Ind_{G,H}, \ind_{G,H} \colon \Mod_R(H) \to \Mod_R(G) $$ respectively denote smooth induction and smooth compactly supported induction.

Section 5.4:

Let $H,K$ be closed subgroups of $G$. $$ I_{K,H} = \Ind_{K,K\cap H}\Res_{K\cap H,H} \text{ and }i_{K,H} = \ind_{K,K\cap H}\Res_{K\cap H,H}. $$

For $\sigma\in \Mod_R(H)$ and $g\in G$, let $g(\sigma)$ denote the representation of $g(H) = gHg^{-1}$ defined by $$ g(\sigma)(ghg^{-1}) = \sigma(h). $$

Section 5.5:

Let $H,K$ be closed subgroups such that all double cosets $HgK$, $g\in G$ are open and closed.

For all $\sigma\in \Mod_R(H)$ we have isomorphisms $$ \Res_{K,G}\Ind_{G,H}(\sigma) \cong \prod_{HgK}I_{K,g(H)}g(\sigma) $$ and $$ \Res_{K,G}\ind_{G,H}(\sigma) \cong \bigoplus_{HgK}i_{K,g(H)}g(\sigma). $$

If $H$ or $K$ is open, then the double cosets are open and closed.

Section 5.6:

Let $H$ be a closed subgroup of $G$.

  1. Let $W\in \Mod_R(H)$ be admissible. Then $\Ind_{G,H}W$ is admissible iff $\ind_{G,H}W$ is admissible, and if they are admissible then $\Ind_{G,H}W = \ind_{G,H}W$.
  2. If $G/H$ is compact, then $\Ind_{G,H}=\ind_{G,H}$ preserves admissibility.

Section 5.7:

Let $H$ be a closed subgroup of $G$.

  1. $\Ind_{G,H}$ is right adjoint to $\Res_{H,G}$.
  2. If $H$ is open, then $\ind_{G,H}$ is left adjoint to $\Res_{H,G}$.
$\endgroup$
1
  • $\begingroup$ Could the downvoter explain their vote, so I know how to improve my answer? :-) $\endgroup$
    – Aurel
    Oct 13, 2019 at 7:34
2
$\begingroup$

The answer to your question is the main theorem of :

Kutzko, P. C. Mackey's theorem for nonunitary representations. Proc. Amer. Math. Soc. 64 (1977), no. 1, 173–175.

You've got to be careful with the induction functors. In the setting of locally profinite groups, the "good" analogue of the induction functor of finite group representations is the compact-induction functor from an open compact subgroup.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.