Contents 1. Introduction 1 2. Classical groups 5 3. Reductive dual pairs 10 4. Weil representations 12 5. Local theta correspondence 13 6. Doubling seesaw 16 7. Degenerate principal series representations 18 8. Normalized intertwining operators 20 9. Doubling zeta integrals 22 10. Standard γ-factors 25 11. Standard γ-factors and local theta correspondence 27 12. Plancherel measures and local theta correspondence 33 13. Formal degrees 37 14. The formal degree conjecture 41 15. Formal degrees and local theta correspondence 45 16. An explicit local theta lifting 50 17. A local Siegel-Weil formula 53 18. A local Rallis inner product formula 55 19. Determination of the constant C — the unramified case 58 20. Determination of the constant C — the equal rank case 63 Appendix A. Analytic properties of normalized intertwining operators 76 Appendix B. Properties of Plancherel measures 81 Appendix C. Convergence of integrals 87 Appendix D. Computation of volumes 91 References 99

Formal degrees and local theta correspondence

By the relative trace formula approach of Jacquet�Rallis, we prove the global Gan�Gross�Prasad conjecture for unitary groups under some local restrictions for the automorphic representations

https://www.math.columbia.edu/~wzhang/math/online/GGP1%20final.pdf

Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups

Let E/F be a quadratic extension of number fields. We study periods and regularized periods of cusp forms and Eisenstein series on \(\operatorname {GL}_{n}( \mathbf {A}_{E})\) over a unitary group of a Hermitian form with respect to E/F. We provide factorization for these periods into locally defined functionals, express these factors in terms of suitably defined local periods and characterize global distinction. We also study in detail the analogous local question and analyze the space of invariant linear forms under a unitary group.

/pdf/on-representations-distinguished-by-unitary-groups-40p94lp0jd.pdf

On representations distinguished by unitary groups

This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data.

Distinguished tame supercuspidal representations

https://www.degruyter.com/view/j/crll.2004.2004.issue-575/crll.2004.080/crll.2004.080.pdf

Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups

This paper analyzes the space HomH(π, 1), where π is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations π which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.

/pdf/tame-supercuspidal-representations-of-gl-n-distinguished-by-uawf0inq6x.pdf

Tame Supercuspidal Representations of GL( n ) Distinguished by a Unitary Group

In this paper we study reducibility of representations of split classical p-adic groups induced from self-contragredient supercuspidal represetation associated via Howe's construction to an admissible character, we show that in many cases Shahidi's criterion for reducibility of the induced representation reduces to a simple condition on the admissible character.

/pdf/reducibility-of-some-induced-representations-of-split-3x8a0xxx3z.pdf

Reducibility of some Induced Representations of Split Classical p-Adic Groups

The topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

Local Character Expansions for Supercuspidal Representations of U(3)

In this paper we study reducibility of those representations of quasi-split unitary p-adic groups which are parabolically induced from supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe’s construction to an admissible character, we show that in many cases a criterion of Goldberg for reducibility of the induced representation reduces to a simple condition on the admissible character.

/pdf/reducibility-of-some-induced-representations-of-p-adic-1p3eraf3t6.pdf

Reducibility of some induced representations of p-adic unitary groups

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