Associated varieties and Howe’sN-spectrum

Degenerate Whittaker functionals for real reductive groups

“…For π ∈ M (G) we define rank(π) = rank(WF π) and depth(π) = depth(WF π). By [He08] rank(π) coincides with the Howe rank of π, and for GL(n), depth (π) coincides with the notion of depth in section 5. It would be interesting to give a representation-theoretic characterization of depth for other classical groups.…”

“…Moreover it suffices to prove the lemma for simple factors of g 0 . The result is obvious for split and complex factors, and by [He08] the other possible simple quasi-split factors are of the form g 0 su l,l su l,l+1 so l,l+2 e 6(2) Label AIII (r = 2l − 1) AIII (r = 2l) DI (r = l + 1) EII Now the lemma can be checked using Table VI of [He08], where (2) means that there are no black dots in the Satake diagram, and (3) means that each of the multiplicities m λ and m 2λ is at most 2.…”

Generalized and degenerate Whittaker quotients and Fourier coefficients

“…Let us add that for irreducible unitary representations of type I classical reductive groups, the Howe rank ( [How82]) is determined by the maximal among the ranks of the matrices lying in the annihilator variety, by [He08].…”

On Equality of Ranks of Local Components of Automorphic Representations

“…4], the algebra H Kn K 0 is isomorphic to the commutant of the image of the group algebra of K n in the induced representation Ind Kn K 0 1, and therefore it is a finite dimensional semisimple associative algebra. Since H K 0 ⊆ S(U ν ), for every smooth representation σ of U ν and every φ ∈ H K 0 we define σ(φ) as in (15).…”

“…Finally, we remark that it would be interesting to see if our main theorem can be used to obtain interesting information about the wavefront sets of the local components of a small automorphic representation of an exceptional group G (see [14] and [15] for results in this direction). We intend to come back to this problem in the near future.…”