Mod-$p$ maximal compact inductions do not have irreducible admissible subrepresentations
Peng Xu
math.RT math.NT 22E50, 11F70





Mod-p maximal compact inductions do not have irreducible admissible subrepresentations
    Peng Xu
    
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Let p be a prime number. We show in this short note that mod-p maximal compact inductions of a p-adic split reductive group do not have irreducible admissible subrepresentations.




§ INTRODUCTION


Let F be a non-archimedean local field of residue characteristic p, and G be a p-adic split reductive group defined over F. Let K be a hyperspecial maximal compact subgroup of G. Let σ be an irreducible smooth 𝐅_p-representation of K. We show in this short note:


The compactly induced representation ind^G _K σ does not have irreducible admissible subrepresentations.










§ PROOF OF THEOREM <REF>



Let σ be an irreducible smooth 𝐅_p-representation of K.



Assume π is an irreducible admissible 𝐅_p-representation of G contained in ind^G _K σ, i.e., we are given a G-embedding


ι: π↪ind^G _K σ.


Take an irreducible smooth 𝐅_p-representation σ_1 of K contained in π. By Frobenius reciprocity, we get a non-zero G-map θ in the space Hom_G (ind^G _K σ_1, π), which will be denoted by H(σ_1, π). Since π is admissible, the space H(σ_1, π) is finite dimensional. By composition, the space H(σ_1, π) is a right module over the spherical Hecke algebra ℋ (K, σ_1): =End_G (ind^G _K σ_1). As G is split, the algebra ℋ (K, σ) is commutative (<cit.>). Therefore we may replace θ by an eigenvector. That is to say, there is a character χ: ℋ (K, σ_1)→𝐅_p so that θ is annihilated by the kernel of χ.



Now under our assumption, the composition ι∘θ is a non-zero map in Hom_G (ind^G _K σ_1, ind^G _K σ). We take a non-zero T in the kernel of χ. We get


(ι∘θ)∘ T =ι∘ (θ∘ T)=0.


As T and ι∘θ are both non-zero, we get a contradiction by the argument of <cit.>.



Note that we have assumed π is admissible in the theorem. However, in certain cases such an assumption is not necessary, say G= GL_2/ U(2, 1), as in both cases a weaker substitute, i.e., the existence of Hecke eigenvalues (<cit.>, <cit.>), is available. Note also that Daniel Le proved recently that there are non-admissible irreducible mod-p smooth representations of GL_2 (𝐐_p^3) (<cit.>).



One interest of Theorem <ref> is to compare it with the complex case: when G is a classical group, a complex maximal compactly induced representation of G might often happen to be irreducible.





amsalpha



