$\DeclareMathOperator\Hom{Hom}$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.