On Equality of Ranks of Local Components of Automorphic Representations

Abstract

We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense of [31]. Our theorem is an analogue of the results previously obtained by Howe [16], Li [21], Dvorsky-Sahi [9], and Kobayashi-Savin [19]. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian unipotent radicals, our proof works uniformly for all of the (classical as well as exceptional) groups under consideration. Our result is an extension of the statement known for several semisimple groups (see [12], [30]) that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal.

Conformally invariant differential operators on Heisenberg groups and minimal representations

“…[14,15]) or the study of local components of global automorphic representations (see e.g. [2,40]). Some of these works use L 2 -realizations of minimal representations.…”

“…Table D.1). The two exceptions are g ≃ g 2 (2) where J ≃ R and n(z) = z 3 and g ≃ sl(n, R) where n(z) = 0 for all z ∈ J . We write (a, z) for aA + z ∈ RA ⊕ J = Λ.…”

“…Remark 6.2.2. As mentioned above, the formula for π min (w 1 ) can be found in [32,Theorem 2] and [48,Proposition 4.2] for the cases G = SO(n, n), E 6( 6) , E 7( 7) , E 8(8) and G 2 (2) . Note that these are all integer cases.…”

“…Recall the elements T 1 , T 2 , T 3 ∈ k from Proposition 5.1.6 and Remark 5.1.7 which span the ideal k 1 ≃ su (2). The element T 2 spans a maximal torus in k 1 and ad( T 2 ) has eigenvalues 0, ±2i.…”

“…Minimal representations are not only of importance in representation theory, but are for instance related to Fourier coefficients of automorphic forms [2,14,15,31], and their realizations often establish connections to families of special functions [24,35] as well as interesting analytic and geometric questions [37,38,39]. It is therefore desirable to have convenient models for these representations.…”