Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)

Kei Yuen Chan

doi:10.1515/crelle-2021-0066

Article

Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)

Kei Yuen Chan

Published/Copyright: December 2, 2021

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 783

Abstract

We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.

A Some homological algebra

Let 𝒜=Alg⁢(Gl). Let ℬ=Alg⁢(Gn). Via Yoneda extension, any element in Ext𝒜1⁢(X,Y) corresponds to a short exact sequence in 𝒜, and zero element corresponds to the split sequence. Then, for an additive exact functor ℱ, ℱ sends a short exact sequence to a short exact sequence, and this defines a map from Ext𝒜1⁢(X,Y) to Extℬ1⁢(ℱ⁢(X),ℱ⁢(Y)).

Lemma A.1.

Let C be a full Serre subcategory of A=Alg⁢(Gl). Let B=Alg⁢(Gn) and let D be a Serre full subcategory of B. Let F:C→D be an exact additive functor. We also regard objects in C as objects in A via the inclusion. Assume that:

any object in 𝒞 is of finite length,
for any simple objects X,Y in the subcategory 𝒞, the induced map of ℱ, from Ext𝒜1⁢(X,Y) to Extℬ1⁢(ℱ⁢(X),ℱ⁢(Y)) is an injection,
ℱ⁢(X) is a simple object in 𝒟 if Xis simple in 𝒞,
for any simple objects X and Y in 𝒞, ℱ⁢(X)≅ℱ⁢(Y) if and only if X≅Y.

Then for any objects X,Y in C, the induced map from ExtA1⁢(X,Y) to ExtB1⁢(F⁢(X),F⁢(Y)) is also injective, and F:C→D is fully-faithful, i.e.

Homℬ⁢(ℱ⁢(X),ℱ⁢(Y))≅Hom𝒟⁢(ℱ⁢(X),ℱ⁢(Y))≅Hom𝒞⁢(X,Y)≅Hom𝒜⁢(X,Y)

for any objects X,Y in C.

Proof.

Let X and Y be objects in 𝒞. When both lengths of X and Y are 1 in 𝒞,

Hom𝒟⁢(ℱ⁢(X),ℱ⁢(Y))≅Hom𝒞⁢(X,Y),Ext𝒜1⁢(X,Y)↪Extℬ1⁢(ℱ⁢(X),ℱ⁢(Y))

are guaranteed by (2), (3) and (4). We first fix the length of X to be at most some n. We shall prove the statement for arbitrary Y by induction on the length of Y.

For an object Y in 𝒞, let Y1 be an irreducible quotient of Y. Then we have a short exact sequence

0→Y2→Y→Y1→0.

Since 𝒞 is Serre, it follows that Y1 and Y2 are in 𝒞.

Note that we have the following commutative diagram:

where the horizontal maps come from long exact sequences, in which the connecting homomorphism is the Yoneda product, and vertical maps for Ext1 are described in the beginning of this section, and the vertical map for Hom is the map induced from the functor.

We have the first vertical arrow is isomorphism and the second and forth vertical arrows are injections by induction hypothesis. Then it is direct to check that the third vertical arrow is also an injection.

Now we consider another commutative diagram:

The first and third vertical arrows are isomorphisms by induction and the last vertical arrow is an injection by induction again. Thus we have that the second vertical arrow is an isomorphism.

Now we switch the role of X and Y, and use similar argument to prove that the assertion is true for X and Y of arbitrary finite length. ∎

Remark A.2.

The above lemma is also valid for arbitrary abelian categories 𝒜 and ℬ which are Schurian k-categories, where k is a field, i.e.

Hom𝒜⁢(X,X)≅k and Homℬ⁢(Y,Y)≅k

for any simple objects X and Y in 𝒜 and ℬ, respectively.

Acknowledgements

This project grows out from discussions with Dipendra Prasad, and the author would like to thank him for helpful discussions and comments. He would also like to thank Gordan Savin for discussions on various topics and helpful comments. The author would also like to thank Max Gurevich for helpful correspondences on the preprint. The author would like to thank the referee for careful reading and useful comments.

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Received: 2020-07-23

Revised: 2021-10-04

Published Online: 2021-12-02

Published in Print: 2022-02-01

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https://doi.org/10.1515/crelle-2021-0066