Generalized Formal Degree

Yannan Qiu
2011 International mathematics research notices  
Let G be a reductive group over a local field of characteristic zero or a finite central cover of such a group. We present a conjecture that enables one to define formal degree for all unitary representations of G. The conjecture is proved for GL n and SL 2 over real and p-adic fields, together with a formal degree relation concerning the local theta correspondence between SL 2 and SO 3 . by guest on April 15, 2011 imrn.oxfordjournals.org Downloaded from The weight factor Δ(g) s makes the above
more » ... weighted integral absolutely convergent when Re(s) 0. Let X π be the space of functionals on V π ⊗ V π ⊗ V π ⊗ V π that is linear in the first and fourth factors, and anti-linear in the second and third factors; then I (s) is an X-valued function well defined on a right half-plane. by guest on April 15, 2011 imrn.oxfordjournals.org Downloaded from Generalized Formal Degree 3 Third, we use the notion of pointwise holomorphicity for one-variable functions valued in X π since X π is without topology. Let F : C → X π be a function; F (s) is called exists a meromorphic function f(s) such that f(s)F (s) is holomorphic; thus I (s) is said to have meromorphic continuation to C if there exists a C-valued meromorphic function f(s) such that f(s)I (s) has holomorphic continuation to C. For an X π -valued meromorphic function F (s), we define its order of zero at a point s 0 by If n 0 is the order of zero at s = s 0 , then we define the leading coefficient of and finite for meromorphic F (s).) Our study of I (s) for representations of classical groups guides us to propose the following conjecture and definition. Conjecture 1.1. Let F be R or a p-adic field, G a finite central cover of a connected reductive group over F , and π an irreducible unitary G-representation. (i) I (s) has meromorphic continuation to C; (ii) The leading coefficient of I (s) at s = 0 is G × G-invariant; (iii) When G is a connected reductive group, let γ (s, π, ad, ψ) = (s, π, ad, ψ) · L(1−s,π ∨ ,ad) L(s,π,ad) be the adjoint γ -factor of π with respect to a nontrivial character ψ of F . Then ord s=0 I (s) = −ord s=0 γ (s, π, ad, ψ). Definition 1.2. Suppose that Conjecture 1.1 is true for a unitary G-representation π . If d π (s) is a meromorphic function and satisfies I st = lim s→0 d π (s)I (s), then we call it a formal degree factor of π. The leading coefficient of a formal degree factor at s = 0 is called the (generalized) formal degree of π and denoted by d(π ). Here we do not explicitly deal with complex reductive groups because reductive groups over C can be naturally considered as reductive groups over R. Before presenting the main results of this paper, we make several conceptual remarks. Remark 1.3. When π is square-integrable, the generalized formal degree coincides with the usual formal degree (cf. Lemma 2.5); when π is nonsquare-integrable, s = 0 may be a pole of I (s) and its order measures the divergence of the integral in the formal by guest on April 15, 2011 imrn.oxfordjournals.org Downloaded from 4 Y. Qiu functional I (−). If s = 0 is indeed a pole of I (s), then the value of the generalized formal degree depends on the concrete choice of the height function; such a dependence should be elementary in nature and we are still in search of a law or a canonical choice of the height function. Remark 1.4. When G is a connected reductive group, part (iii) of Conjecture 1.1 implies the existence of a constant d 0 (π ) such that d 0 (π )γ (s, π, ad, ψ) is a formal degree factor. d 0 (π ) should be of arithmetic nature; the Hiraga-Ichino-Ikeda conjecture (cf. [8, Conjecture 1.4]) is essentially a formula of d 0 (π ) for square-integrable representations and we expect a general but similar formula working for all unitary representations. Remark 1.5. The generalized formal degree has the potential to relate to the Plancherel measure. Suppose that the center of G is anisotropic and (π, V π ) is a nonsquareintegrable tempered unitary representation of G, then π belongs to a family {π t } t∈T , where the index set T is a compact real torus and the representations π t are actions on V π . Let dt be a Haar measure on T and λ(t) dt be the Plancherel measure on T; let v be a unit vector in V π and define the v-Fourier transform for continuous functions on T by If K(s) has meromorphic continuation, then the isometry of the v-Fourier transform implies T ϕ 1 (t)ϕ 2 (t)λ(t) dt = Gφ 1 (g)φ 2 (g) dg = lim s→0 T×T ϕ 1 (t 1 )ϕ 2 (t 2 )λ(t 1 )λ(t 2 )K(s; t 1 , t 2 ) dt 1 dt 2 . In other words, when s → 0, one has K(s) → λ(t) −1 1 ΔT in the sense of distribution, where ΔT is the diagonal of T × T. A closer look at the limiting behavior of K(s)| ΔT , that is,
doi:10.1093/imrn/rnr015 fatcat:6ebzhr7ogjdahozmy6c4vi4odi