In this note for p > 5 we calculate the dimensions of Ext 1 SL2(Qp) (τ, σ) for any two irreducible supersingular representations τ and σ of SL 2 (Q p ).
Let F be any non-Archimedean local field with residue field of cardinality {q_{F}} . In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form {[\operatorname{GL}_{n}(F)\times F^{\times},\sigma\otimes\chi]} with {q_{F}>2} , and for the principal series components with {q_{F}>3} . With this we complete the classification of typical representations for {\operatorname{GL}_{3}(F)} , for {q_{F}>2} .
Let g = g0 ⊕ g1 be a Z2-grading of a classical Lie algebra such that (g, g0) is a classical symmetric pair. Let G be a classical group with Lie algebra g and let G0 be the connected subgroup of G with Lie(G0) = g0. For d ≥ 2, let C d (g1) be the d-th commuting scheme associated with the symmetric pair (g, g0). In this article, we study the categorical quotient C d (g1)//G0 via the Chevalley restriction map. As a consequence we show that the categorical quotient scheme C d (g1)//G0 is normal and reduced. As a part of the proof, we describe a generating set for the algebra k[g d 1 ] G 0 , which are of independent interest.
Let 𝔤 {\mathfrak{g}} be a complex semisimple Lie algebra and let θ be a finite-order automorphism of 𝔤 {\mathfrak{g}} . Let 𝔤 0 {\mathfrak{g}_{0}} be the subalgebra { X ∈ 𝔤 : θ ( X ) = X } {\{X\in\mathfrak{g}:\theta(X)=X\}} . In this article, we study for which pairs ( V 1 , V 2 ) {(V_{1},V_{2})} , consisting of two irreducible finite-dimensional representations of 𝔤 {\mathfrak{g}} , we have res 𝔤 0 V 1 ≃ res 𝔤 0 V 2 . \operatorname{res}_{\mathfrak{g}_{0}}V_{1}\simeq\operatorname{res}_{\mathfrak{% g}_{0}}V_{2}. In many cases, we show that V 1 {V_{1}} and V 2 {V_{2}} have isomorphic restrictions to 𝔤 0 {\mathfrak{g}_{0}} if and only if V 1 {V_{1}} is isomorphic to V 2 σ {V_{2}^{\sigma}} for some outer automorphism σ of 𝔤 {\mathfrak{g}} .
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