Character Theory

Jeffrey Adams University of Maryland

Slides: www.math.umd.edu/ ˜ jda/preprints Introduction

Representation Theory of Semisimple Lie Groups

A representation π of a group G is a complicated object, often infinite dimensional. The character of π, which determines π, is a function on G.

Unexpected and deep relationships between different groups can be seen by looking at the characters of their representations. It is often much harder to see the corresponding relationship between the representations.

“Characters Matter”

Motivation Two themes in number theory:

• Trace Formula (Langlands Program) - relate representations of algebraic (i.e. matrix) groups H, G.

Basic problem: compute the multiplicity of a representation in the space of automorphic forms. This is very hard. “Endoscopy” compares the multiplicities for representations of G with those for other simpler groups H.

• Theta-Correspondence - relate representations of possibly non-algebraic groups G_{1}, G_{2}.

Quantum mechanics: The quantum harmonic oscillator gives rise to the oscillator representation of the non-algebraic two-fold cover of SL(2, ℝ), more generally Sp(2n, ℝ). This gives rise to relationships between representations of subgroups (G_{1}, G_{2}) of S̃p(2n, ℝ).

Problem: Bring the Theta-Correspondence into the Langlands Program.

Approach: Character Theory - Relate the representation theory of each non-algebraic group to an algebraic group, using character theory.

Finite Groups

G = finite group GL(n, ℂ) = {n × n complex matrices g | det(g) ≠ 0} π : G → GL(n, ℂ) a representation (group homomorphism) Theorem: (Schur) π is determined (up to isomorphism) by its character, the function

Θ_{π}(g) = Trace (π(g))

Corollary: (Pointwise Conjugacy ⇒ Conjugacy) Suppose ι_{1} and ι_{2} are two embeddings of G in GL(n, ℂ), satisfying:

ι_{1}(g) ∼ ι_{2}(g)

for all g ∈ G. Then there exists x ∈ GL(n, ℂ) such that for all g ∈ G,

xι_{1}(g)x^{−1} = ι_{2}(g)

(In other words, if for all g ∈ G there exists x_{g} ∈ GL(n, ℂ) such that x_{g}ι_{1}(g)x_{g}^{−1} = ι_{2}(g), then x_{g} can be chosen independent of g.) Proof: The assumption implies

⇒ Θ_{ι_{1}} = Θ_{ι_{2}}
⇒ ι_{1} ≃ ι_{2} (Theorem )
⇒ xι_{1}(g)x^{−1} = ι_{2}(g) (g ∈ G)

by definition of isomorphism.

Remark: This result is a special feature of GL(n, ℂ). It fails with GL(n, ℂ) replaced by other groups such as P GL(n, ℂ) (n ≥ 3). This is related to the failure of multiplicity one for SL(n) [Blasius 1994].

The character Θ_{π} is conjugation invariant:

Θ_{π}(xgx^{−1}) = Θ_{π}(g)

Theorem: (Frobenius) {Θ_{π} | π ∈ G } is a basis of L^{2}(G)^{G}, the conjugation invariant functions on G

Note that {χ_{C} | C a conjugacy class } is also a basis of L^{2}(G)^{G}. The matrix relating these two bases is the character table of G. For example:

δ_{1} = ∑ ; π∈Gdim (π)Θ_{π}

Equivalently, as a representation of G×G (acting by left and right translation),

L^{2}(G) ≃ ∑ ; π∈Gπ ⊗ π^{∗}

Reductive Groups

G=reductive Lie group over 𝔽 real or p-adic, for example GL(n, 𝔽), Sp(2n, 𝔽)

G_{0}: the (strongly) regular semi-simple elements. These elements are diagonalizable over an algebraic closure, and G_{0} is an open and dense subset of G.

Theorem: (HC 1965, 1978) Let π be an irreducible unitary (more generally: admissible) representation of G.

• The character Θ_{π} of π is defined as a distribution: Θ_{π}(f) = T r(π(f)) for f smooth and compactly supported,

• Θ_{π}(f) = ∫_{G_{0}} f(g)Θ_{π}(g)dg where Θ_{π} is a function on G_{0},

• The function Θ_{π} is independent of the choice of Haar measure dg, and Θ_{π} determines π up to isomorphism.

All of the complicated structure of an infinite dimensional representation π is therefore encoded in a single function Θ_{π}. This function can be very complicated, and carries deep number theoretic information.

Suppose 𝔽 = ℝ. By analogy with finite groups, the “discrete” part of L^{2}(G) may be written

L_{d}^{2}(G) = ∑ π ⊗ π^{∗}

The (countable set of) π which appear are the “discrete series” of G. In spite of this natural definition, it is very difficult to construct π directly. Harish Chandra first classified the discrete series by studying their characters [HC 1965′].

The formulas for the characters of the discrete series are very complicated. They have been computed independently in [Herb 1983], [GKM 1997], and in some special cases in [Zuckerman 1976?], [Schmid 1974].

Exercise: Write a computer program to compute these formulas. Compare them for G of type E_{8}.

Example: (Kazhdan) The character formula of certain discrete series representation of p-adic groups involve counting the number of points on an elliptic curve over a finite field.

Although the general character of a representation (even a finite dimensional representation of a compact group) may be complicated, certain very special representations have remarkable characters.

Example: Let G = Spin(n), the two-fold cover of SO(n). The Spin representation of G, of dimension 2^{[\frac{n}{2}]}, has character

|Θ_{Spin}(g)| = \frac{1}{√2^{c}}|det(1 + g)|^{\frac{1}{2}}

(c = 0 or 1 respectively if n is even or odd.) This is simpler than the character of a general finite-dimensional representation of G.

Remark: The spin representation is the smallest genuine representation of Spin(n) (genuine: does not factor to SO(n)).

Example: Let G = S̃p(2n), the two-fold cover of Sp(2n) over a real or p-adic field. Let

ω = ω_{+} ⊕ ω_{−}

be the oscillator representation, and

Θ_{ω} = Θ_{ω_{+}} ⊕ Θ_{ω_{−}}

its character. Then [Howe]

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

The phase is easily determined up to a factor τ(g) with τ(g)^{8} = 1. The calculation of τ(g) involves some deep questions number theory.

Example: Let G, ω, Θ_{ω} be as in the preceding example, with 𝔽 now a finite field of order q. Then [Howe 1973]

Θ_{ω}(g) = τ(g)q^{\frac{1}{2} dim (Ker(g−1))}

with τ(g)^{4} = 1

τ(g) = Γ(Q_{g}, ψ).

Here Γ is the standard Gauss sum associated to a certain quadratic from Q_{g} on Ker(g − 1).

Remark: The oscillator representation is the smallest nontrivial unitary representation of S̃p(2n, ℝ).

Problem: Compute the character of “small” unitary representations, for example the “minimal” representation. At least in some cases these should be simpler than the character of an arbitrary representation.

Small representations often occur only for non-algebraic groups.

Basic Principle

Suppose the semisimple conjugacy classes of two Lie groups G, H are related in some simple way. Since the character of a representation is a function on conjugacy classes, the representation theory of G, H should be related, by carrying over the characters.

More precisely, suppose there is a map

Ψ : H_{0}/ ∼→ G_{0}/ ∼

Write Ψ : C_{H} → C_{G} for the map a conjugacy classes. Given a representation π_{H} of H, with character Θ_{π}, define:

Θ_{G}(C_{G}) = ∑_{C_{H},Ψ(C_{H})=C_{G}} Φ(C_{H}, C_{G})Θ_{π_{H}}(C_{H}) (1)

Here Φ(C_{H}, C_{G}) is a factor having to do with the difference between measures on H, G.

The question is then: is Θ_{G} the character of a representation π_{G} of G? If so, study the correspondence π_{H} → π_{G}.

Example: If H ⊂ G, Θ_{G} is the character of the induced representation Ind_{H}^{G}(π).

Non-algebraic groups

The Langlands program deals with an algebraic group G, i.e. G is a closed subgroup of a matrix group GL(n).

The theta correspondence (and its generalizations) involves non-algebraic groups. The existence of non-algebraic covers of Sp(2n) is implied by quantum mechanics. Even the existence of non-algebraic covers of other groups is a deep subject, intimately related to the theory of reciprocity laws in class field theory [Moore 1968].

If G is an algebraic group over a local field 𝔽, H^{2}(G) (in the appropriate category) parametrizes the central extensions of G. H^{2}(G) is almost entirely independent of G: in most cases H^{2}(G) ≃ μ(𝔽) (the roots of unity in 𝔽). Brylinski and Deligne have recently given a very general construction of these central extensions [BD 1998].

The representation theory of these non-algebraic groups is somewhat mysterious. Flicker, Kazhdan and Patterson studied related the character theory of cover to GL(n) to GL(n) itself [Flicker 1980], [FK 1986], [KP 1984]. This is the first case of program outlined at the beginning of the talk, and of (1) for non-algebraic groups.

It is difficult to see the corresponding map on representations directly. For example, it takes the trivial representation of GL(2) to the oscillator representation of the two-fold cover of GL(2).

Example: Let H = Spin(2n + 1) over ℝ (the split spin group), G = S̃p(2n, ℝ). Define a correspondence between semisimple conjugacy classes of SO(2n+1) and Sp(2n, ℝ) as follows: τ(g) = g′ if g, g′ have the same non-trivial eigenvalues. This is a bijection between SO(2n+1, ℂ) and Sp(2n, ℂ) orbits of (strongly) regular semisimple elements.

Proposition: There is a unique lifting τ̃ of τ to a correspondence between Spin(2n + 1) and S̃p(2n, ℝ), such that if τ(g) = g′ then

Θ_{Spin}(g)Θ_{ω_{+} −ω_{−}}(g′) = \frac{1}{√2}

This is an aspect of Boson-Fermion duality: the spin representation is on the exterior algebra Λ(V), and the oscillator representation is on the symmetric algebra S(W) ≃ S(V). This is also related to Koszul duality.

S̃p(2n)-SO(2n + 1) duality

𝔽 :local, of characteristic zero, i.e. ℝ, ℂ or p-adic V, (,): symmetric bilinear form W, <, >: skew-symmetric (symplectic) bilinear form 𝕎 = V ⊗ W, <<, >>, skew-symmetric bilinear form 𝕎 ≃ Hom_{𝔽}(V, W), G = SO(V), G′ = Sp(W), 𝔤 = Lie(G), 𝔤′ = Lie(G′),

Hom (V, W) → Hom (W, V) :
T → T^{∗}
< w, T v > = (T^{∗}w, v)

Lemma: T^{∗}T ∈ Λ^{2}(V) ≃ 𝔤 ⊂ Hom (V, V), T T^{∗} ∈ S^{2}(W) ≃ 𝔤′ ⊂ Hom (W, W), This defines the orbit correspondence between G-orbits on 𝔤 and G′-orbits on 𝔤′.

Remark: (T T^{∗})^{m} = 0 implies (T^{∗}T)^{m+1} = 0, so this defines a correspondence of nilpotent orbits.

There are only finitely many nilpotent orbits, and they form a cone.

Problem: Study the correspondence of nilpotent orbits given by the remark in more detail. It should have some interesting consequences in geometry and representation theory.

For characters we are mainly interested in the semisimple orbits:

Remark: T T^{∗}v = λv implies (T^{∗}T)(T^{∗}v) = λT^{∗}v, so T T^{∗} and T^{∗}T have the same non-zero eigenvalues.

In fact:

Proposition: Fix W and δ ∈ 𝔽^{∗}/𝔽^{∗2}. Let

V = {(V, (,)) | dim (V) = dim (W) + 1, disc (V) = δ}/ ∼

Then the orbit correspondence is a bijection:

Sp(W)_{0}/ ∼↔ ⋃_{V ∈V} SO(V)_{0}/ ∼

This very strong relationship of conjugacy classes suggests a strong relationship between the representations.

However, there is no natural relationship between the representations of Sp(W) and SO(V). The reason: the oscillator representation.

Remark: The representations of SO(V) may be obtained in a simple manner from the representations of the split form SO(V_{0}) ≃ SO(n + 1, n) (at least conjecturally). So for the application to representations it is enough to work with just the split form on the right hand side.

Let p : S̃p(W) → Sp(W) be the metaplectic group, with oscillator representation ω as before, and let

Φ = Θ_{ω_{+}} − Θ_{ω_{−}},

as a function on S̃p(2n)_{0}.

Let V be the split orthogonal space with dim (V) = dim (W)+ 1. Write

g ↔ g′ (g ∈ SO(V), g′ ∈ Sp(W))

for the orbit correspondence, and for g′ ∈ S̃p(W), write g ↔ g′ if g ↔ p(g′).

For π a genuine representation of SO(V), with character Θ_{π}, define:

Θ′(g′) = Φ(g′)Θ_{π}(g) (g ↔ g′)

This is a conjugation invariant function on S̃p(W).

This is a special case of 1.

Conjecture: Suppose π is a stable representation, i.e. its character is invariant by conjugation by SO(2n+1, \overline{𝔽}). Then

(a) Θ′ is the character of a stable representation π′.

(b) π → π′ is a bijection:

S̃p(W) _{genuine,stable} ⟷ SO(V) _{stable}

This reduces the study of genuine representations of S̃p(W) to those of SO(V).

Theorem: (Adams 1998) The conjecture is true for 𝔽 = ℝ.

Theorem: (Schultz 1998) The conjecture is true if 𝔽 padic and n = 1.

The case of n = 1 is related to work of Shintani, Waldspurger and others on modular forms of half-integral weight.

Bibliography

[Adams 1998] J. Adams, Lifting of Characters on Orthogonal and Metaplectic Groups, Duke Math. J., Vol. 92, no. 1, pp. 129– 178 (1998).

[Blasius 1994] D. Blasius, On multiplicities for SL(n), Israel J. Math. 88 (1994), no. 1-3, 237–251.

[BD 1998] J.-L. Brylinski, P. Deligne, Central Extensions of Reductive Groups by K_{2}, preprint.

[Flicker] Y. Flicker, Automorphic Forms on Covering Groups of GL(2), Invent. Math. 57 (1980), 119-182.

[FK 1986] Y. Flicker, D. Kazhdan, Metaplectic Correspondence Proc. Inst. des Hautes Etudes Sci., 64, (1986) pp. 53–110

[GKM 1997] M. Goresky, R. Kottwitz, R. Macpherson, Discrete series characters and the Lefschetz formula for Hecke operators. Duke Math. J. 89 (1997), no. 3, pp. 477–554.

[HC 1965] Harish-Chandra, Invariant Eigendistributions on a Semisimple Lie Groups, Trans. AMS, 119 (1965), 457–508.

[HC 1965′] Harish-Chandra, Discrete series for semisimple Lie groups I and II, Acta Math. Vol. 113 (1965), pp. 241-318; Vol. 116 (1966), pp. 1-111.

[HC 1978] Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, Queen’s Papers in Pure and Applied Math., 48 (1978), 281–347.

[Herb 1983] R. Herb, Discrete series characters and Fourier inversion on semisimple real Lie groups. Trans. Amer. Math. Soc. 277 (1983), no. 1, pp. 241–262.

[Howe] R. Howe, Appendix, Notes on Dual Pairs, Yale University, preprint.

[Howe 1973] R. Howe, On the Character of Weil’s representation, Trans. AMS, 177 (1973), 287–298.

[KP 1986] D. Kazhdan, S. J. Patterson, Metaplectic Forms Proc. Inst. des Hautes Etudes Sci., 59 (1984)

[Moore 1968] C. Moore, Group extensions of p-adic and adelic linear groups. Inst. Hautes Etudes Sci. Publ. Math. No. 35 (1968) pp. 157–222.

[Schmid 1974] W. Schmid, Some remarks about the discrete series characters of Sp(n,ℝ). Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), pp. 172–194.

[Schultz 1998] J. Schultz, Thesis, University of Maryland, 1998

[Zuckerman 1976?] G. Zuckerman, Thesis, Princeton University.