(August 15, 2008)
Characters of principal series Paul Garrett
[email protected] http://www.math.umn.edu/˜garrett/
The characters of principal series representations of reductive groups G over local fields, whether p-adic or archimedean, admit a simple common computation. In particular, these characters have smaller support than other (irreducible) representations, such as discrete series, or representations induced from discrete series. The archimedean case was known to Gelfand-Graev, HarishChandra, and others in the early 1950’s. The book of Gelfand-Graev-PiatetskiShapiro on GL2 emphasizes the commonalities of GL2 ( ) and GL2 ( p ). Jacquet’s work also made clear that the commonality of p-adic and archimedean cases extends to larger reductive groups.
R
Q
Let P be a minimal parabolic subgroup of G, and χ a one-dimensional complex representation of P factoring through the quotient P/N of P by its unipotent radical N . Let M ≈ P/N be a Levi component of P . Let Iχnf be the naively normalized induced representation constructed as [1]
C
Iχnf = IndG P χ = { -valued functions f on G : f (pg) = χ(p) · f (g)} with G acting by right translations. Note that, by an Iwasawa decomposition G = P · K, the values of a function f ∈ Iχnf are determined by its values on K. Functions η ∈ Cco (G) act as usual on any representation π of G by superpositions [2] Z Z η · v = π(η)(v) = η(g) g · v dg = η(g) π(g)(v) dg G
G
On π = Iχ , with x ∈ K, Z (η · f )(x) =
Z η(g) f (xg) dg =
G
Z Z = K
P
η(x−1 py) f (py)
η(x−1 g) f (g) dg
G
dp dy = δ(p)
Z Z K
η(x−1 py) χ(p) f (y)
P
dp dy δ(p)
where δ is the modular function of P , and dp is a right Haar measure, so dp/δ(p) is a left Haar measure. This operator can be described by a kernel: extract the inner integral over P by setting Z dp Qη (x, y) = η(x−1 py) χ(p) δ(p) P so then
Z (η · f )(x) =
Qη (x, y) f (y) dy K
[1] We should require two further things in the description of the induced representation. First, we should require
that the functions in the induced representation restrict to L2 functions on a maximal compact K. This makes the 1 induced representation a Hilbert space. Second, we should require that |χ/δ 2 | = 1, where δ is the modular function of P . That is, we should require that this renormalized version of χ is unitary. This condition assures that that right action of G is unitary, as can be verified by changing variables. [2] These integrals of operators are Gelfand-Pettis integrals of compactly supported continuous operator-valued
functions. The topology on the space of operators is the strong operator topology given by seminorms |T |v = |T v| as v ranges over the Hilbert space. This topology is strictly coarser than the uniform-norm operator topology. Indeed, this strong topology is not metrizable, but is quasi-complete and locally convex, so Gelfand-Pettis integrals exist and enjoy their usual properties.
1
Paul Garrett: Characters of principal series (August 15, 2008) The integral for Qη (x, y) has support inside K · spt(η) · K, which is compact because the support of η is compact. By Gelfand-Pettis Qη (x, y) is continuous on K×K. Thus, the operator f → η·f is Hilbert-Schmidt. If it were of trace class, it would have trace computed by Z Z Z dp dx tr (f → η · f ) = Qη (x, x) dx = η(x−1 px) χ(p) δ(p) K K P To make the symmetries of the latter integral more apparent, reparametrize the copy of P via the map n × m → n−1 mn
N × M −→ P
The image includes all semi-simple elements of P , and is injective on the regular elements M reg of M , essentially by definition. This map is N -equivariant, so the change-of-measure can be determined near 1 × m. From n−1 mn = n−1 · mnm−1 · m ∈ N · M the change-of-measure is Y α(m) − 1
∆(m) =
(for m ∈ M reg )
α>0
where the product is over positive roots α, with multiplicities. Thus, Z Z Z tr (f → η · f ) = η(x−1 (n−1 mn) x) χ(n−1 mn) ∆(m) K
Z Z Z =
−1
η((nx) K
N
M reg
N
M reg
dm dx = m (nx)) χ(m) ∆(m) dn δ(m)
Z M \G
Z
dn dm dx δ(n−1 mn)
η(g˙ −1 m g) ˙ χ(m)
M reg
∆(m) dm dg˙ δ(m)
Observe that ∆/δ is almost invariant under conjugation of M by the Weyl group W , but we are off by a 1 1 factor of δ 2 . Thus, replacing the naive χ by the better-normalized δ 2 · χ, Y 1 Y 1 1 |α 2 − α− 2 | |α − 1| = χ · δ 2 · χ · δ −1 · α>0
α>0
if we ignore the issue of ambiguity of individual square roots of characters. Since the totality of roots consists of positive roots and negatives of positive roots, and since the Weyl group preserves the negation of roots, the latter product is W -stable. Thus, we should use the less naive normalization Iχ = Iδnf1/2 ·χ Thus, summing over conjugates of the integrand by W , on the less-naively normalized Iχ , Z Z 1 ∆(m) X w χ (m) dm dg˙ η(g˙ −1 m g) ˙ 1 tr (f → η · f ) = #W M \G M reg δ 2 (m) w∈W
Z = M reg
∆(m) X 1
δ 2 (m)
χw (m)
Z
η(g˙ −1 m g) ˙ dg˙ dm
M \G
w∈W
Without worrying further about normalizations, this expresses the trace in terms of an orbital integral Z OηM (m) = η(g˙ −1 m g) ˙ dg˙ M \G
of η, namely Z tr (f → η · f ) = M reg
∆(m) X 1 2
δ (m)
2
w∈W
χw (m) · OηM (m) dm
