Mackey theory in the setting of locally profinite groups

$$\DeclareMathOperator\Hom{Hom}$$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.

• You'll probably have a lot better luck if you state, or at least summarise, the theorem that you want generalised. Oct 11, 2019 at 17:41
• Crosspost of math.stackexchange.com/questions/3388557
– YCor
Oct 12, 2019 at 9:28
• 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 $(x, y)$? Oct 12, 2019 at 21:54
• @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 $x^{-1}y \in D$. Oct 13, 2019 at 0:25


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 $$\Res_{H,G}$$.
• Could the downvoter explain their vote, so I know how to improve my answer? :-) Oct 13, 2019 at 7:34