Characters of Non-Linear Groups

Notes available at www.math.umd.edu/ ˜ jda/preprints. \underline{General setup}: 𝔽: a local field of characteristic 0 𝔾: an algebraic group over 𝔽, often simply connected, G = 𝔾(𝔽), G̃ a central extension of G:

1 ⟶ A ⟶ G̃ ⟶^{p} G ⟶ 1

(|A| < ∞). When 𝔾 is simply connected G̃ is a non-linear group. Relevant semisimple orbits Definition: An irreducible representation π of G̃ is genuine if the central character χ_{π} is an injection when restricted to A. Fix a Cartan subgroup T of G, let T̃ = p^{−1}(T), Z(T̃) the center of T̃. Definition: If g ∈ T̃ is regular,

g is relevant if g ∈ Z(T̃)

The next Proposition originates in [Flicker]: Proposition: Suppose π is an irreducible genuine representation, with character Θ_{π}. If g is not relevant then

Θ_{π}(g) = 0

Proof: Suppose g is not relevant. Then

g ∼_{T̃} g′

for some g′ ≠ g. Projecting both sides to G gives p(g) ∼_{T} p(g′) ⇒ p(g) = p(g′) ⇒ g′ = ag for some a ∈ A, a ≠ 1. Therefore

g ∼ g′ = ga

Since Θ_{π} is conjugation invariant this gives

Θ_{π}(g) = Θ_{π}(ga) = χ_{π}(a)Θ_{π}(g).

Since π is genuine, χ_{π}(a) ≠ 1 proving the result.

Typeset by AMS-TEX

Remark: Rebecca Herb noticed an error in one version of Harish-Chandra’s character formula for the discrete series. In [Acta 1965] formula (pg. 304) is correct, but in (Lemma 57) the terms are rearranged, and written:

Θ_{π}(g) = \frac{1}{Δ(g)} ∑_{t∈w\W_{G}} ϵ(t)λ^{t}(h_{1}) ∑_{s∈W (A^{+})} ϵ(s)c_{λ}(s : t : A^{+})exp(st(λ)^{y}(H_{2}))

The first sum is not well defined for g ∉ Z(T). However by the preceding discussion this may be corrected as follows:

Θ_{π}(g) = \{ \begin{matrix}\frac{1}{Δ(g)} ∑_{t∈w\W_{G}} ϵ(t)λ^{t}(h_{1}) \\ ∑_{s∈W (A^{+})} ϵ(s)c_{λ}(s : t : A^{+})exp(st(λ)^{y}(H_{2})) & g ∈ Z(T) \\ 0 & g ∉ Z(T) \end{matrix}

Remark: A similar phenomenon holds for G̃ a linear cover of G. Recall g is regular if Cent_{G}(g)^{0} is a torus, and strongly regular if Cent_{G}(g) is a torus. Suppose g ∈ G̃ and p(g) is regular, but not strongly regular. Then Θ_{π}(g) = 0 for any genuine representation of G̃.

Example: G = SO(2n), G̃ = Spin(2n). Suppose g ∈ SO(2n) is regular. Then g is not strongly regular if and only both ±1 are eigenvalues of g.

|Θ_{Spin_{+}} ± Θ_{Spin_{−}}(g)| = |det(1 ± p(g))|^{\frac{1}{2}}

It follows that Θ_{Spin_{±}}(g) = 0 if and only if p(g) is not strongly regular.

Spin-Oscillator Duality

Let G = Sp(2n, ℝ), G′ = SO(n + 1, n). For g, g′ semisimple elements, write g ↔ g′ if g, g′ have the same eigenvalues (This is the stabilized orbit correspondence.)

Let osc = osc_{+} ⊕ osc_{−} be the oscillator representation of S̃p(2n, ℝ). Then (Howe)

|Θ_{osc}(g)| = |det(1 − p(g))|^{− \frac{1}{2}}

Proposition: The correspondence g → g′ may be lifted uniquely to a correspondence g̃ → g̃′ between S̃p(2n, ℝ) and Spin(n + 1, n) with the following property:

Θ_{osc_{+}−osc_{−}}(g̃)Θ_{Spin}(g̃′) = 1

Remark: In absolute value this follows from

|Θ_{Spin}(g)| =|det(1 + p(g))|^{\frac{1}{2}}
|Θ_{osc_{+}±osc_{−}}(g)| =|det(1 ∓ p(g))|^{− \frac{1}{2}}

Flicker/Kazhdan/Patterson Lifting

Let G = GL(n, 𝔽), G̃ an N-fold central extension.

Flicker (for GL(2)) and Kazhdan/Patterson (for GL(n)) defined an operation t_{∗}, taking a virtual character of G to a genuine virtual character of G̃. If σ is irreducible and tempered, then t_{∗}(σ) is zero or ± an irreducible tempered representation. Over ℂ (Tadic) and ℝ (Adams-Huang), for σ irreducible unitary, t_{∗}(σ) is zero, or ± an irreducible unitary representation.

\underline{Inversion}

Flicker/Kazhdan/Patterson lifting completely describe the irreducible characters of covers of GL(n). What about { \atop SL(n)}?

Consider a non-trivial N − fold cover { \atop SL(n)} of SL(n), and assume N divides n, for example the 2-fold cover of SL(2). This extends to a cover { \atop GL(n)} of GL(n) (c = 0 in Kazhdan/Patterson’s notation).

Let π be an irreducible genuine representation of { \atop GL(n)}, and write ∑_{i} π_{i} for the

restriction of π to { \atop SL(n)}.

Let

GL(n)_{+} = SL(n)D = {g ∈ GL(n) | det(g) ∈ 𝔽^{∗n}}

The key point is:

Z({ \atop GL(n)_{+}})/Z({ \atop GL(n)}) ≃ 𝔽^{∗}/𝔽^{∗N}

Write χ_{i} for the central character of π_{i} considered as a representation of { \atop GL(n)_{+}}. Choose representatives x_{i} of 𝔽^{∗}/𝔽^{∗N}, and z_{i} ∈ Z({ \atop GL(n)_{+}}), p(z_{i}) = x_{i}I.

Proposition:

θ_{π_{i}}(g) = \frac{1}{m} ∑_{j} χ_{i}(z_{j})^{−1}θ_{π}(z_{j}g)

Remark: This is very different from the case of linear groups, for which the character of a representation of SL(n) is not obtained directly from the corresponding characters of GL(n). In fact for SL(2) this is the first example of endoscopy, in which an endoscopic group T (an elliptic torus) plays a role.

Support of Distributions

It is interesting to consider the restriction to S̃L(n) of invariant distributions of { \atop SL(n)}. By lifting a virtual character of GL(n) which vanishes on SL(n), we obtain a virtual character of SL(n) with small support. To obtain such a virtual character of GL(n), simply take π − π ⊗ β for some character β.

For simplicity we consider the case GL(2), N = 2. (The general situation is similar, up to complications arising from the center of G̃L(n)). Suppose σ is a virtual representation of GL(2), and let π = t_{∗}(σ). By [Flicker],

Θ_{π}(g) = ∑_{\begin{matrix}h \\ h^{2}=p(g) \end{matrix}} Φ(h)Θ_{σ}(h)

Here Φ(h) is a (transfer) factor which need not concern us here. Note that if p(g) = h^{2} is elliptic, the sum is over ±h.

Now let σ be an irreducible representation of GL(2), with t_{∗}(σ) ≠ 0. Fix a character β_{0} of 𝔽^{∗} with β_{0}(−1) = −1, and let β(g) = β_{0}(det(g)). Let π = t_{∗}(σ − σ ⊗ β), restricted to S̃L(2).

Suppose g is elliptic. By the preceding formula, Θ_{π}(g) = 0 if

(1) p(g) = h^{2} with h ∈ SL(2),

(2) p(g) ≠ h^{2} for any h ∈ GL(2)

Therefore: the support of Θ_{π}, restricted to the regular elliptic set in S̃L(2), is

{g | p(g) = h^{2}, det(h) = −1}

This is a fairly small but non-empty set. Suppose T is an elliptic torus in GL(2), isomorphic to the multiplicative group 𝔼^{∗} of a quadratic extension 𝔼 = 𝔽(\sqrt{Δ}). The determinant of an element z ∈ 𝔼^{∗} is the norm N(z), the torus in SL(2) is the norm one elements, and the set we are considering is

{z^{2} | N(z) = −1}

This is non-empty if and only if (−1, Δ)_{𝔽} = 1. For example if −1 is not a square in 𝔽 and the residual characteristic is odd, this holds if and only if Δ = −1. In this case Θ_{π} is supported on:

{g ∈ T − T^{2} | T unramified}.

Note that T − T^{2} = −T^{2} is a coset of index 2 in T.

Finally we note that π may be zero. By considering the hyperbolic set, we see π ≠ 0

if Θ_{σ}(g) ≠ 0 for some g = ( \begin{matrix}x \\ & −\frac{1}{x} \end{matrix} ).

For example, taking σ to be the trivial representation gives pi is the sum of the oscillator representations of S̃L(2) (there are |𝔽^{∗}/𝔽^{∗2}| such representations).