Discrete Series and Characters of the Component Group

Jeffrey Adams

April 9, 2007

Suppose ϕ : W_{ℝ} → ^{L}G is an L-homomorphism. There is a close relationship between the L-packet associated to ϕ and characters the component group of the centralizer of ϕ. This is reinterpreted in [2], see also [1], in part to make it more canonical and a bijection. This involves a number of changes, including using the notion of strong real form, several strong real forms at once, and taking a cover of the component group. This cover is not necessarily a two-group, so the values of the character may not be just signs.

Over the years a number of people have asked me how to relate an Lpacket of discrete series to characters of S_{ϕ} in this language. While this is a special case of [2] and [1], it isn’t so easy to extract it. In these notes I work out this case, and give some details in the case of SU(p, q) and U(p, q). There are no proofs; see the references for more details.

Besides the basic references cited above, I make some use of [?].

If you want to cut to the chase the main result is Proposition 4.4. Also see Propositions 3.3, 7.3 and 8.2.

1 Basic Setup

We fix a pair (G, γ) consisting of a reductive, algebraic (or complex) group and an outer automorphism of G. This corresponds to an inner class of real forms of G. We are interested in discrete series representations of real forms of G. A real form in this inner class has discrete series representations if and only if γ = 1, so we assume this.

The extended group is a semidirect product G^{Γ} = G ⋊ Γ where Γ = Gal(ℂ/ℝ). Since γ = 1 it is a direct product, and we may safely drop Γ from the notation. The constructions of [2],[1],[?] involving the “twist” simplify in this setting.

We fix Cartan and Borel subgroups H and B of G. Let G^{∨} be the dual group, and fix Cartan and Borel subgroups H^{∨} and B^{∨} of G^{∨}. By construction we have canonical identifications

(1.1)  X^{∗}(H) = X_{∗}(H^{∨}), X_{∗}(H) = X^{∗}(H^{∨})

where X^{∗} and X_{∗} denote the character and co-character lattices, respectively. Let R(G, H), P (G, H), R^{∨}(G, H) and P^{∨}(G, H) be the root, weight, coroot, and coweight lattices of G with respect to H. Recall

(1.2) P = {λ ∈ X^{∗}(H) ⊗ ℚ | ⟨λ, α^{∨}⟩ ∈ ℤ for all α^{∨} ∈ R^{∨}} P^{∨} = {γ^{∨} ∈ X_{∗}(H) ⊗ ℚ | ⟨α, γ^{∨}⟩ ∈ ℤ for all α ∈ R}

and these are lattices if and only if G is semisimple. We abbreviate these R, P, R^{∨} and P^{∨}. Define R(G^{∨}, H^{∨}) etc. similarly. Then, for example R(G, H) = R^{∨}(G^{∨}, H^{∨}). Let W = W (G, H) be the Weyl group.

Write G^{∨Γ}, instead of ^{L}G, for the L-group of G. This is a semidirect product G^{∨} ⋊ Γ.

2 Strong real forms

A strong real form of G is an element x ∈ G satisfying x^{2} ∈ Z (Z is the center of G) (recall we are assuming γ = 1), and two such elements are said to be equivalent if they are conjugate by G. A real form of G is an involution θ ∈ Int(G) (this is the Cartan involution), and equivalence is conjugation by G. The map x → int(x) defines a surjection

(2.1)  {strong real forms of G}/ ∼→ {real forms of G}/ ∼ .

If G is adjoint this is a bijection.

Every strong real form x is conjugate to an element of H. Let

(2.2)  X_{1} = {h ∈ H | h^{2} ∈ Z(G)}.

If G is semisimple this is a finite set. The Weyl group W acts on X_{1}, and there is a bijection

(2.3)  {strong real forms of G}/ ∼⟷^{1−1} X_{1}/W.

Let X′_{1} be a set of representatives of X_{1}/W.

For γ^{∨} ∈ P^{∨} let x(γ) = exp(πiγ^{∨}) ∈ X_{1} and z(γ^{∨}) = exp(2πiγ^{∨}) ∈ Z. This gives an isomorphism X_{1} ≃ P^{∨}/2X_{∗}.

The element x(ρ^{∨}) = exp(πiρ^{∨}) plays a special role (ρ^{∨} is one-half the sum of the positive co-roots). The corresponding real form is quasisplit. The element z(ρ^{∨}) = x(ρ^{∨})^{2} is independent of the choice of positive roots.

It is often convenient consider only those x with x^{2} fixed. So fix z ∈ Z and let

(2.4)  X_{1}[z] = {x ∈ H | x^{2} = z}, X′_{1}[z] = {x ∈ X′_{1} | x^{2} = z}.

3 L-parameters

Suppose ϕ is an admissible homomorphism of the Weil group of ℝ into G^{∨Γ}. Associated to x ∈ X_{1} and ϕ is a finite set Π(x, ϕ), the L-packet for the strong real form x corresponding to ϕ. This may be empty. We have Π(x, ϕ) = Π(x′, ϕ) if and only if x is equivalent to x′, i.e. x′ = wx for some w ∈ W. Define

(3.1)  Π(ϕ) = ∐_{x∈X′_{1}} Π(x, ϕ),

the L-packet associated to ϕ.

Unless G is adjoint these sets are often larger than necessary. For one thing if G is not semisimple they are infinite. Secondly, multiple strong real forms corresponding to the same real form can occur. See the examples.

For these reasons it is sometimes helpful to fix an element z ∈ Z, and define

(3.2)  Π(z, ϕ) = ∐_{x∈X′_{1}[x]} Π(x, ϕ).

(If we think of z ∈ Z as an element of Z or X_{1} then Π(z, ϕ) has two possible meanings; hopefully this will not cause any confusion.)

These sets are finite, and the number of strong real forms mapping to single real form is small, and often 1. On the other hand for a fixed z some real forms may fail to occur. See the examples.

Let λ be the infinitesimal character for G determined by ϕ. For the purposes of this discussion this infinitesimal character is not important, and we can take it to be ρ, the infinitesimal character of the trivial representation.

Proposition 3.3 Suppose ϕ is a discrete series L-parameter for G. There are natural bijections

(3.4)  X_{1} ⟷^{1−1} Π(ϕ) = ∐_{x∈X′_{1}} Π(x, ϕ)

and

(3.5)  X_{1}[z] ⟷^{1−1} Π(z, ϕ) = ∐_{x∈X′_{1}[z]} Π(x, ϕ).

This proposition practically proves itself. Write the infinitesimal character corresponding to ϕ as λ ∈ 𝔥^{∗}, dominant with respect to the fixed choice of Borel subgroup B. Associated to x ∈ X_{1} is a real form with Cartan involution θ_{x} = int(x) and (complexified) maximal compact subgroup K_{x} = Cent_{G}(x). Associated to x is a discrete series representation, which we denote π_{x}(λ) of this real form, with Harish-Chandra parameter λ. The key point is that if x′ = wx with w ∈ W, there is a natural way to identify the representation π_{wx}(λ) with π_{x}(w^{−1}λ). See [?].

4 Characters of S_{ϕ}

Fix a discrete series L-homomorphism ϕ and let S_{ϕ} = Cent_{G^{∨}}(ϕ). This is a two-group. The theory of Langlands and Shelstad relates the elements of Π(ϕ) to characters of S_{ϕ}.

In [3] S_{ϕ} is replaced by a certain cover of it, denoted S̃_{ϕ}, and defines a canonical bijection between Π(ϕ) and S̃_{ϕ} . Unlike S_{ϕ}, however, S̃_{ϕ} is not a two-group, and if G is not semisimple is not even finite. This version of the theory has the advantage that it is canonical, but it is also somewhat unsatisfying, and harder to relate to the classical theory.

For this reason we fix an element z ∈ Z, and work with X_{1}[z] ≃ Π(z, ϕ). Fix x_{0} ∈ X_{1}[z]. The map x → xx_{0}^{−1} defines an isomorphism

(4.1)  X_{1}[z] ≃ H_{2} ≃ X_{∗}/2X_{∗},

the elements of order 2 in H.

The lack of a canonical choice of x_{0} ∈ X_{1}[z] is problematic. The case of z = z(ρ^{∨}) is an important one, for which there is a canonical such choice. In this case we can take x_{0} = x(ρ^{∨}) = exp(πiρ^{∨}). The corresponding (strong) real form is quasisplit, and x_{0} in the bijection of Proposition 3.3 x_{0} will corresponds to a generic discrete series representation of the quasisplit form.

The strong real forms x ∈ X_{1}[z(ρ^{∨})] are called pure [4]. (In [4] these are given by x^{2} = 1; the difference is due to the fact that we defined G^{Γ} with respect to the maximally compact real form, instead of the quasisplit one.) Not every real form is represented by a pure strong real form. For such real forms a choice is necessary. See the examples.

Let X_{1,pure} = X_{1}[z(ρ^{∨})] and X′_{1,pure} = X′_{1}[z(ρ^{∨})].

Now consider S_{ϕ}. Without loss of generality we may assume ϕ(ℂ^{×}) ⊂ H^{∨}. Then ϕ(j) normalizes H^{∨} and (since this is a discrete series parameter) acts on H^{∨} by h → h^{−1}. Then it is easy to see

(4.2)  S_{ϕ} = H_{2}^{∨} ≃ X_{∗}(H^{∨})/2X_{∗}(H^{∨}) ≃ X^{∗}/2X^{∗}.

Write s(μ) for the element of S_{ϕ} corresponding to μ ∈ X^{∗}.

For x ∈ X_{1} let χ_{x} be the corresponding character of S_{ϕ}, i.e.

(4.3)  χ_{x}(s(μ)) = μ(x) (μ ∈ X^{∗}).

The natural pairing between these X_{∗}/2X∗ and X^{∗}/2X^{∗} gives:

Proposition 4.4 Suppose ϕ is a discrete series L-parameter. For x ∈ X_{1,pure} let

(4.5)  τ_{x} = χ_{xx(ρ^{∨})^{−1}} = χ_{x}χ_{x(ρ^{∨})}^{−1}.

Then the map x → τ_{x} induces a canonical bijection

(4.6)  ∐_{{x∈X′_{1,pure}}} Π(x, ϕ) ⟷^{1−1} Ŝ_{ϕ}.

In this bijection the trivial character of S_{ϕ} corresonds to a generic discrete series representation of a quasisplit (strong) real form.

More generally fix z ∈ Z and x_{0} ∈ X_{1}[z]. The map

(4.7)  x → τ_{x} = χ_{x}χ_{x_{0}}^{−1} = χ_{xx_{0}^{−1}}

induces a bijection

(4.8)  ∐_{{x∈X′_{1}[z]}} Π(x, ϕ) ⟷^{1−1} Ŝ_{ϕ}

depending on the choice of x_{0}.

The left hand side of (4.6) is the disjoint union, over various strong real forms of G, of a classical L-packet for the corresponding real form. For any real form of G in this inner class there may be a pure strong real form mapping to it, and if so there may be more than one. The pure strong real form x(ρ) is quasisplit.

Every real form occurs in (4.8) for some z.

We make these maps explicit. For x ∈ X_{1,pure} we have

(4.9)  τ_{x}(μ) = μ(xx(ρ^{∨})^{−1}) (μ ∈ X^{∗}/2X^{∗}).

More generally given x_{0} ∈ X_{1}[z] we have

(4.10)  τ_{x}(μ) = μ(xx_{0}^{−1}) (μ ∈ X^{∗}/2X^{∗}).

Alternatively write x_{0} = x(γ_{0}^{∨}); recall γ_{0}^{∨} = ρ^{∨} in the case of pure strong real forms. Then

(4.11)  χ_{x(γ^{∨})}(s(μ)) = (−1)^{⟨μ,γ^{∨}−γ_{0}^{∨}⟩}

5 Example: Strong Real forms of SU(p, q)

Let n = p + q.

First assume n is odd. Then z(ρ^{∨}) = I and x = diag(±1, …, ±1) with an even number of minus signs. There is a bijection between real forms and strong real forms. For example if n = 5 we have SU(5, 0), SU(3, 2) and SU(1, 4).

Suppose n = 2m is even. Then z(ρ^{∨}) = −I, and

(5.1)  X_{1}[z(ρ^{∨})] = diag(ϵ_{1}i, …, ϵ_{n}i)

with ϵ_{i} = ±1 and ∏ ϵ_{i} = (−1)^{m}. In particular x(ρ^{∨}) = diag(\overbrace{i, …, i}^{m}, \overbrace{−i, …, −i}^{m}) and this corresponds to the quasisplit form SU(m, m). The strong real forms in X_{1,pure} correspond to real forms SU(p, q) with p ≡ q (mod 2); the non-quasisplit ones each occur twice. If m is even we can think of these as SU(2m, 0), SU(2m − 2, 2), … SU(0, 2m). If m is odd we have SU(2m − 1, 1), SU(2m − 3, 3), …, SU(1, 2m − 1).

For example if n = 2m = 4 we can think of the strong real forms as SU(4, 0), SU(2, 2) and SU(0, 4). If n = 2m = 6 we get SU(5, 1), SU(3, 3) and SU(1, 5).

In this case not every real form occurs. To get all real forms we must choose another element z. The most convenient such choice is z = I if m is odd, or z = iI if m is even.

Suppose m is odd. Let z = I. Then the elements x = diag(±1, …, ±1) with and even number of minus signs give the groups SU(2m, 0), SU(2m − 2),…, SU(0, 2m); each real form occurs twice. For example for n = 2m = 6 we have SU(6, 0), SU(4, 2), SU(2, 4) and SU(0, 6).

Now assume m is even. Then take z = iI and let α = e^{πi/4}. Then x = α(±1, …, ±1) with an odd number of minus signs. We get each SU(p, q) with p, q odd counted twice. For example if n = 2m = 8 we have SU(7, 1), SU(5, 3), SU(3, 5) and SU(1, 7).

The following table summarizes the situation. The number of groups in the row labelled by z is the number of inequivalent strong real forms corresponding to z. The occurence of two groups SU(p, q) and SU(q, p) means that the map from strong real forms to this real forms is two to one. If only one of these groups occurs it is one to one.

1. n = 2

(a) z = z(ρ^{∨}) = −I: SU(1, 1)

(b) z = I: SU(2, 0), SU(0, 2)

2. n = 3

(a) z = z(ρ^{∨}) = I: SU(3, 0), SU(1, 2)

3. n = 4:

(a) z = z(ρ^{∨}) = −I: SU(4, 0), SU(2, 2), SU(0, 4)

(b) z = iI: SU(3, 1), SU(1, 3)

4. n = 5:

(a) z = z(ρ^{∨}) = I: SU(5, 0), SU(3, 2), SU(1, 4)

5. n = 6:

(a) z = z(ρ^{∨}) = −I: SU(5, 1), SU(3, 3), SU(1, 5),

(b) z = I: SU(6, 0), SU(4, 2), SU(2, 4), SU(0, 6)

It isn’t entirely necessary, but to be precise below we specify names for strong real forms of SL(n, ℂ). Let α = 1, i or e^{πi/4}, and let SU(p, q) be the strong real form defined by

(5.2)  x = diag(\overbrace{α, …, α}^{p}, \overbrace{−α, …, −α}^{q}).

(assuming this has determinant 1).

6 Example: Strong Real Forms of U(p, q)

Since G is not semisimple the theory is complicated if we allow all strong real forms. On the other hand if we fix z = z(ρ^{∨}) the situation is quite simple.

Let n = p + q. We have z(ρ^{∨}) = I if n is odd, and −I if n is even.

Suppose n is odd. Then x = diag(±1, …, ±1) with any number of minus signs. Write U(p, q) for the strong real form defined by x where 1 is an eigenvalue of multiplicity p and −1 has multiplicity q. Note that the strong real forms U(p, q) and U(q, p) are not equivalent; they map to the same real form.

Suppose n is even. Then z(ρ^{∨}) = −I, and we take x = i(±1, …, ±1). Write U(p, q) for the strong real form defined by x with i^{n−1} having multiplicity p and −(i^{n−1}) having multiplicity q. (The exponent isn’t critical, and is included for the sake of Proposition 7.3). If p ≠ q the strong real forms U(p, q) and U(q, p) are inequivalent, and both map to the same real form, which is therefore counted twice. There is unique quasisplit strong real form denoted U(m, m) mapping to the quasisplit real form.

In each case the strong real forms are labelled U(n, 0), U(n−1, 1), …, U(0, n).

7 Discrete series of U(p, q)

Let G = GL(n, C) and suppose ϕ is a discrete series L-parameter. Then S_{ϕ} ≃ ℤ/2ℤ^{n}, and Ŝ_{ϕ} is in bijection with a set of discrete series representations of various strong real forms of G. We make this more explicit.

We fix z = z(ρ^{∨}) = (−1)^{n+1}I. Let

(7.1)  \begin{aligned}z_{0} = z(ρ^{∨}) & = (−1)^{n−1}I \\ x_{0} = x(ρ^{∨}) & = i^{n−1}diag(1, −1, …, (−1)^{n−1}) \end{aligned}

Write S_{ϕ} = {diag(±1, …, ±1)}. For ϵ_{i} = ±1 let χ(ϵ_{1}, …, ϵ_{n}) be the corresponding character of S_{ϕ}. Then χ(ϵ_{1}, …, ϵ_{n}) = χ_{x} with

(7.2) x = diag(ϵ_{1}, …, ϵ_{n})x_{0} = i^{n−1}diag(ϵ_{1}, −ϵ_{2}, ϵ_{3}, −ϵ_{4}, …, (−1)^{n−1}ϵ_{n}).

The upshot is this.

Proposition 7.3 Let G = GL(n, ℂ) and suppose ϕ is a discrete series Lparameter for G. There is a canonical bijection between Ŝ_{ϕ} and the discrete series representations of the strong real forms {U(p, q) | 0 ≤ p, q ≤ n, p + q = n}.

Fix a character χ(ϵ_{1}, …, ϵ_{n}) and let

(7.4)  v = (ϵ_{1}, −ϵ_{2}, ϵ_{3}, −ϵ_{4}, …, (−1)^{n−1}ϵ_{n}).

Then χ corresponds to a representation of the strong real form U(p, q) where p is the number of 1′s in v, and q is the number of −1′s.

For example the trivial character goes to the quasisplit strong real form U(m, m) or U(m + 1, m).

Note that each real form U(p, q) occurs twice, as the strong real forms U(p, q) and U(q, p), unless p = q. The total number of discrete series representation of U(n, 0), U(n − 1, 1), …, U(0, n) is 2^{n} = |S_{ϕ}|.

8 Discrete series of SU(p, q)

This case is a bit more complicated than U(p, q). Write S_{ϕ} = {diag(±1, …, ±1)} with and even number of minus signs, and write χ(ϵ_{1}, …, ϵ_{n}) accordingly.

If n is odd things are reasonably straightforward. Recall (Section 5) the map from strong real forms to real forms is a bijection.

In this case

(8.1)  \begin{aligned}z(ρ^{∨}) & = I \\ x(ρ^{∨}) & = (−1)^{\frac{n−1}{2}} diag(1, −1, …, 1). \end{aligned}

Proposition 8.2 Let G = SL(n, ℂ) with n odd. There is a canonical bijection between Ŝ_{ϕ} and the discrete series representations of the real forms, equivalently strong real forms,

(8.3)  {SU(p, q) | 0 ≤ p, q ≤ n, p + q = n, q even}.

Let

(8.4)  v = (−1)^{\frac{n−1}{2}} (ϵ_{1}, −ϵ_{2}, ϵ_{3}, …, −ϵ_{n−1}, ϵ_{n}).

Then χ corresponds to the real form SU(p, q) where p is number of 1′s in v, and q the number of −1′s.

Suppose n is even. Recall not every real form occurs in X_{1,pure} in this case, and the map from strong real forms to real forms is not bijective.

Proposition 8.5 Let G = SL(n, ℂ) with n = 2m. Let z_{0} = z(ρ^{∨}). There is a canonical bijection between Ŝ_{ϕ} and the discrete series representation of the strong real forms

(8.6)  \{SU(p, q) | 0 ≤ p, q ≤ n, p + q = n, p ≡ q ≡ \frac{n}{2} (mod 2)\}.

These are the pure strong real forms.

Let

(8.7)  v = (ϵ_{1}, −ϵ_{2}, ϵ_{3}, …, ϵ_{n−1}, −ϵ_{n}).

Then χ corresponds to the real form SU(p, q) where p is number of 1′s in v, and q the number of −1′s.

If m is even let α = e^{πi/4} and z′_{0} = iI. If m is odd let α = 1 and z′_{0} = I. Choose

(8.8)  x_{0} = αdiag(δ_{1}, …, δ_{n})

with δ_{i} = ±1. Then x_{0}^{2} = z_{0}, and there is a bijection between Ŝ_{ϕ} and the discrete series representations of the strong real forms

(8.9)  \{SU(p, q) | 0 ≤ p, q ≤ n, p + q = n, p ≡ q ≡ \frac{n}{2} + 1 (mod 2)\}.

Let

(8.10)  v = (δ_{1}ϵ_{1}, …, δ_{n}ϵ_{n}).

Then χ corresponds to the real form SU(p, q) where p is number of 1′s in v, and q the number of −1′s.

We mention just one example of the group S̃_{ϕ}. Let G = SL(2, ℂ). Then S_{ϕ} = ℤ/2ℤ and S̃_{ϕ} = ℤ/4ℤ. There is a natural bijection between the discrete series representations of the strong real forms SU(2, 0), SU(0, 2) and SU(1, 1) and characters of S̃_{ϕ}.

The two characters of S̃_{ϕ} of order 2 factor to S_{ϕ}; these correspond to the two discrete series representation of the (pure) strong real form SU(1, 1) as before. The trivial representations of SU(2, 0) and SU(0, 2) correspond to the two characters of S̃_{ϕ} of order 4. Via the non-canonical choice of (8.10) these also correspond to the two characters of S̃_{ϕ} of order 2, or equivalently the two characters of S_{ϕ}.

References

[1] J. Adams. Lifting of characters, volume 101 of Progress in mathematics. Birkhäuser, Boston,Basel,Berlin, 1991.

[2] Jeffrey Adams, Dan Barbasch, and David A. Vogan, Jr. The Langlands classification and irreducible characters for real reductive groups, volume 104 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1992.

[3] D. Vogan J. Adams. L-groups, projective representations, and the Langlands classification. 113:45–138, 1992.

[4] David A. Vogan, Jr. The local Langlands conjecture. In Representation theory of groups and algebras, pages 305–379. Amer. Math. Soc., Providence, RI, 1993.