BRANCHING LAWS

AIM LECTURES JULY

2003 BY PETER TRAPA

Notes by David A. Vogan, Jr.

July 30, 2003

Abstract. A basic problem in the representation theory of a compact Lie group is to calculate the restriction of an irreducible representation of K to a closed subgroup. Most classical results on this problem concern connected groups. We’ll recall some of those, and talk about some of the modifications needed to work with disconnected groups.

1. Weight multiplicities

Suppose 𝔤 is a complex semisimple Lie algebra and 𝔥 a Cartan subalgebra. If we fix a Borel subalgebra 𝔟 = 𝔥 + 𝔫 of 𝔤, then the irreducible finite-dimensional representations of 𝔤 are parametrized by dominant integral weights 𝔥^{∗}. Write V^{λ} for the representation of highest weight λ. The restriction of V^{λ} to 𝔥 is a direct sum of weight spaces:

V^{λ}|_{𝔥} = ⨁_{μin𝔥^{∗}} m_{λμ}ℂ_{μ},  (1.1)

with m_{λμ} a non-negative integer (the weight multiplicity). The most basic classical branching problem is to calculate these multiplicities. One solution is Kostant’s multiplicity formula:

m_{λμ} = ∑_{w∈W}(−1)^{w}P_{𝔫}(w(λ + ρ) − (μ + ρ)).  (1.2)(a)

Here is what the terms on the right mean. (−1)^{w} is the determinant of the action of w on 𝔥^{∗} (which is ±1). The function

P_{𝔫}: 𝔥^{∗} → ℕ  (1.2)(b)

is Kostant’s partition function: its value at γ is the number of distinct expressions

γ = ∑_{α∈Δ(𝔫,𝔥)} n_{α}α

Typeset by AMS-TEX

of γ as a non-negative integer combination of positive roots. There are many proofs of Kostant’s formula, using for example the BGG resolution of V^{λ}, or the smoothness of the flag variety. A very algebraic reference is [Hump].

Knowledge of weight multiplicities leads directly to solutions of many other branching problems. An example of a problem that it does not solve directly is this: calculate the representation of W on the zero weight space of V^{λ}. The heart of the difficulty is that the group W (or the normalizer of H in G) is disconnected.

Before moving to other topics, notice that two basic ingredients in Kostant’s formula are the Cartan-Weyl parametrization of representations by dominant weights, and the action of W on weights.

2. K multiplicities.

Suppose G is a real reductive group with maximal compact subgroup K and complexified Lie algebra 𝔤. If X is a (𝔤, K)-module of finite length, then by analogy with (1.1) we have

X|_{K} = ⨁_{λ∈K̂} m_{X}^{λ}E^{λ},  (2.1)

with m_{X}^{λ} a non-negative integer. The fundamental branching problem in this setting is computation of m_{X}^{λ}. Here is a way to do that. Suppose for definiteness that X is irreducible. Write Par for the (finite) set of Langlands parameters for representations of infinitesimal character equal to that of X. To each δ ∈ Par there is associated a “standard Harish-Chandra module”

X(δ) = Ind_{MAN}^{G}(X_{ds}^{M} ⊗ e^{ν} ⊗ 1).  (2.2)(a)

Here X_{ds}^{M} is a Harish-Chandra module for M in the limits of the discrete series, and ν ∈ 𝔞^{∗}. Each such standard module has a unique Langlands subquotient \overline{X}(δ). (Uniqueness of the Langlands subquotient depends on choosing exactly the right parameter set; it is not true for arbitrary representations of the form (2.2)(a).) The irreducible modules \overline{X}(δ) are inequivalent, and every irreducible Harish-Chandra module of the same infinitesimal character as X is equivalent to one of them.

The Kazhdan-Lusztig conjectures (which are theorems at least for connected linear groups) allow us to write

\overline{X}(γ) = ∑_{δ∈Par} m_{γδ}X(δ),  (2.2)(b)

with m_{γδ} an integer given by a reasonable^{1} algorithm (see the talk by Fokko du Cloux).

The problem of computing X|_{K} is reduced by (2.2)(b) to the problem of computing each X(δ)|_{K}. Because of (2.2)(a),

X(δ)|_{K} = Ind_{M∩K}^{K}(X_{ds}^{M}|_{M∩K}).

Because of Frobenius reciprocity, this calculation in turn can be broken into two parts:

(1) compute X_{ds}^{M}|_{M∩K}; and

(2) compute V^{λ}|_{M∩K} for every irreducible representation λ ∈ K̂.

1The definition of “reasonable” shifted rather drastically on the occasion of the publication of the original article of Kazhdan and Lusztig.

The first of these problems can be approached using the Blattner formula, which is closely analogous to (and as computable as) the Kostant multiplicity formula (1.2). There are complications arising from the disconnectedness of M, but never mind. We will therefore concentrate on the second, which is entirely a branching problem for compact groups. Here are two illuminating examples.

Example 2.3. Suppose G = U(p, q), so that K = U(p) × U(q). The choices for M are parametrized by a non-negative integer a less than or equal to both p and q. Then

M ∩ K = U(p − a) × U(1)_{Δ}^{a} × U(q − a).

This is the subgroup of [U(p − a) × U(1)^{a}] × [U(1)^{a} × U(q − a)] obtained by taking the diagonal copy of U(1) in each of the a copies of U(1) × U(1). Branching to these subgroups is easily handled by mild generalizations of Kostant’s multiplicity formula.

Example 2.4. Suppose G = O(p, q), so that K = O(p) × O(q). Some of the possibilities for M (there are more) are parametrized by a non-negative integer a less than or equal to both p and q. Then

M ∩ K = O(p − a) × O(1)_{Δ}^{a} × O(q − a).

Formally this example looks very similar to the preceding one, but technically it is very different: the subgroup M ∩ K has (usually) 2^{a+2} connected components, so that one cannot expect to get a branching law using only Lie algebra ideas.

Example 2.5. Suppose G is the split real linear group of type E_{8}, so that K = Spin(16)/{1, ϵ}. If MAN is the Borel subgroup of G, then M = M ∩ K ≃ (ℤ/2ℤ)^{8}. At most 16 of the 256 elements of M belong to a common maximal torus of K, so again Lie algebra methods cannot provide much help with the branching problem.

3. Branching for connected compact groups.

In this section we will recall Kostant’s general branching theorem for connected compact Lie groups. We can work entirely with Lie algebras; so fix a complex reductive Lie algebra 𝔤, and let 𝔨 be a subalgebra reductive in 𝔤. (The connection with compact groups appears when 𝔤 and 𝔨 are complexified Lie algebras for compact connected groups G ⊃ K.) There is an ad(𝔨)-invariant complement 𝔭 for 𝔨:

𝔤 = 𝔨 ⊕ 𝔭.  (3.1)(a)

Choose a Cartan subalgebra 𝔱 ⊂ 𝔨, and extend it to a Cartan subalgebra 𝔥 of 𝔤:

𝔥 = 𝔱 ⊕ (𝔥 ∩ 𝔭).  (3.1)(b)

There is a natural restriction map

res: 𝔥^{∗} → 𝔱^{∗},  λ ↦ \overline{λ}.  (3.1)(c)

Define

𝔩 = 𝔤^{𝔱},  (3.1)(d)

a Levi subalgebra of 𝔤 (corresponding to the roots of 𝔥 in 𝔤 that restrict to zero on 𝔱). Choose a parabolic subalgebra 𝔮 = 𝔩 + 𝔲 of 𝔤 in such a way if α and β are any two roots of 𝔥 in 𝔤, then

\overline{α} = \overline{β} and α ∈ Δ(𝔲, 𝔥) ⇒ β ∈ Δ(𝔲, 𝔥).  (3.1)(e).

This is certainly possible to arrange. Finally, choose

Δ^{+}(𝔤, 𝔥) ⊃ Δ(𝔲, 𝔥).  (3.1)(f).

We need a little notation about the Weyl group. For w ∈ W (𝔤, 𝔥) = W, define^{2}

Δ_{w} = {α ∈ Δ^{+} | w^{−1}α < 0}.  (3.2)(a)

Now define

W′ = {w ∈ W | Δ_{w} ⊂ Δ(𝔲, 𝔥).  (3.2)(b)

The set W′ is the set of (minimal length) coset representatives for W (𝔩, 𝔥) in W; that is, multiplication defines a bijection

W (𝔤, 𝔥) = W (𝔩, 𝔥) × W′.  (3.2)(c)

Theorem 3.3 (Kostant). Suppose V^{λ} is a finite-dimensional irreducible representation of 𝔤 of highest weight λ ∈ 𝔥^{∗}, and V^{ν} is a finite-dimensional irreducible representation of 𝔨 of highest weight ν ∈ 𝔱^{∗}. Then the multiplicity of V^{ν} in V^{λ}|_{𝔨} is

∑_{w∈W′}(−1)^{w}d_{λ}(w)P_{𝔲∩𝔭}(\overline{w(λ + ρ) − ρ} − ν).

Here d_{λ}(w) is the dimension of the irreducible representation of 𝔩 of highest weight w(λ + ρ) − ρ; and the partition function

P_{𝔲∩𝔭}: 𝔱^{∗} → ℕ

counts expressions in terms of the weights of 𝔱 on 𝔲 ∩ 𝔭.

Another approach to this branching problem is due to Patera, Sharp, Moody, and McKay. The idea is to begin with the weight multiplicity formula

V^{λ} = ∑_{μ∈𝔥^{∗}dominant} m_{μ}(W · μ),

then to compute the branching

(W · μ)|_{𝔱} = ∑_{μ′∈𝔱^{∗}dominant} m_{μ′}(W_{𝔨}) · μ′,

and then to deduce multiplicities of representations of 𝔨.

2In the lecture, w was σ, and Δ_{w} was written W_{σ}. This notetaker was able to write that down, but finds himself unable to commit it to TEX.

4. Disconnected compact groups.

We mentioned at the end of section 1 that two of the basic ingredients in the Kostant multiplicity formula are the Cartan-Weyl parametrization of K̂ by weights, and the action of the Weyl group on the parameter space. Here we will begin to extend these ingredients to disconnected groups.

So assume that K is a compact Lie group, possibly disconnected. Write 𝔨 for the complexified Lie algebra of K, T_{0} for a maximal torus in the identity component K_{0}, and

𝔟 = 𝔱 + 𝔫  (4.1)

for a Borel subalgebra of 𝔨.

Definition 4.2. The large Cartan subgroup of K corresponding to 𝔟 is

T^{+} = {t ∈ K | Ad(t)(𝔟 = 𝔟} = N_{K}(𝔟).

The small Cartan subgroup is

T^{−} = {t ∈ K | Ad(t)|𝔱 is trivial} = Z_{K}(T_{0}),

the centralizer in K of T_{0}. A general Cartan subgroup is any group T such that

T^{−} ⊂ T ⊂ T^{+}.

If T is any Cartan subgroup of K corresponding to 𝔟, then there is a well-defined character 2ρ of T,

2ρ(t) = det Ad(t)|_{𝔫},  (4.3)(a)

the determinant of the adjoint action of T on 𝔫. Despite the suggestive notation, the character 2ρ need not have a square root. We need such a square root, so we introduce the group

T̃ = ρ cover of T = {(t, z) ∈ T × ℂ^{×} | 2ρ(t) = z^{2}}.  (4.3)(b)

Projection on the first factor defines a two-to-one group homomorphism

π: T̃ → T,  π(t, z) = t.  (4.3)(c)

The kernel of π consists of the identity and ϵ = (1, −1). There is a short exact sequence

1 ⟶ {1, ϵ} ⟶ T̃⟶^{π}T ⟶ 1.  (4.3)(d)

Projection on the second factor defines a character that we call ρ:

ρ: T̃ → ℂ^{×},  ρ(t, z) = z.  (4.3)(e)

The differential of ρ is equal to one half the differential of 2ρ, or one half the sum of the roots of 𝔱 in 𝔫. Furthermore ρ is a genuine character of T̃, by which we mean that ρ(ϵ) = −1.

Definition 4.4. Suppose that T̃ is the ρ cover of T defined in (4.3). An irreducible representation χ of T̃ is called genuine if χ(ϵ) = −I. (This is equivalent to assuming that χ does not factor through π to the quotient group T.) The differential of χ is a completely reducible representation of the Lie algebra 𝔱, and therefore a direct sum of various weights in 𝔱^{∗}. The representation χ is called dominant if one (or, equivalentally, all) of these weights is dominant. It is called regular if one (or, equivalently, all) of these weights is non-zero on every coroot of 𝔱 in 𝔨.

Theorem 4.5. Suppose K is a compact Lie group and T^{+} is a large Cartan subgroup. Let T̃^{+} be the ρ covering of T^{+} (cf. (4.3)). Then there is a natural bijection between the set K̂ of irreducible representations of K, and the set of genuine dominant regular irreducible representations of T̃^{+}.

Describing this bijection is fairly easy. If V is an irreducible representation of K, then the subspace V^{𝔫} of highest weight vectors is an irreducible representation of T^{+}. This provides a bijection

K̂ ↔ dominant irreducible representations of T^{+}.

Next, ρ is a genuine irreducible one-dimensional representation of T̃^{+}. It follows that tensoring with ρ provides a bijection

irreducible representations of T^{+} ↔ genuine irreducible representations of T̃^{+}.

This bijection identifies dominant representations on the left with dominant regular representations on the right. Combining these two bijections gives the theorem.

You may wonder why we introduced the covering, which so far has served only to complicate the statement and proof of Theorem 4.5. The reason is that the covering will make possible a statement of the Weyl character formula, in which both numerator and denominator are genuine functions on T̃^{+}. (A “genuine function” is one sent to its negative under translation by ϵ.) The actual lecture concluded with a precise and correct statement of the Weyl character formula on the small Cartan subgroup T^{−}. The notes will conclude instead with a statement of the Weyl character formula on T^{+} more or less equivalent to the one described in [Vorange].

Definition 4.6. Suppose K is a compact Lie group, 𝔟 is a Borel subalgebra of 𝔨, and T^{+} is the normalizer of 𝔟 in 𝔨. Then the homogeneous space K/T^{+} is naturally identified with the flag variety B of all Borel subalgebras of 𝔨. An element of k ∈ K is called regular for K if some (equivalently every? I’m not sure what to guess) fixed point of k on B is regular: that is, if the induced action of k on the tangent space at the fixed point does not have 1 as an eigenvalue.

An element t ∈ T^{+} is called regular for K if the adjoint action Ad(t)|_{𝔫} does not have 1 as an eigenvalue. (In order for this definition to be nicely consistent with the one above, one needs to be able to replace “some” by “every.”) Regularity in T^{+} is equivalent to the condition

det(I − Ad(t)|_{𝔫}) ≠ 0.

The Weyl denominator is the function

Δ(t̃) = ρ(t̃)^{−1} det(I − Ad(π(t̃))|_{𝔫})

on T̃^{+}. It is non-zero exactly on the preimages of regular elements of T^{+}.

The (unchecked) hope is that regular elements in T^{+} are dense in T^{+}, and that regular elements of K are dense in K, and that every conjugacy class in K meets T^{+}. None of this should be very hard; the danger is that it’s false.

We turn now to the Weyl character formula. Just as for connected K, the formula expresses the trace of a regular element t ∈ T^{+} as a quotient of two terms, each of which depends on a preimage of t in T̃^{+}. (Changing the chosen preimage changes the sign of each term, so the quotient depends only on t.) The denominator is the Weyl denominator of Definition 4.6. The numerator is (roughly speaking) a sum over the Weyl group of T_{0} in K_{0}. What makes life complicated is that this Weyl group does not act on T^{+}. The solution (roughly speaking) is simply to throw away the terms that don’t make sense.

Here are the details. Recall that we are fixing 𝔟 = 𝔱+𝔫. Write 𝔫^{−} for the opposite nil radical (which is automatically preserved by T^{+}). Suppose w ∈ W (𝔨, 𝔱). Then w defines a second Borel subalgebra

𝔟^{w} = 𝔱 + 𝔫^{w},  (4.7)(a)

corresponding to w(Δ^{+}(𝔨, 𝔱). This new Borel subalgebra defines a large Cartan subgroup

T^{+,w} = N_{K}(𝔟^{w}).  (4.7)(b)

The identity component of T^{+,w} is the same torus T_{0} as for T^{+}, but the other components can be different. It’s not difficult to see that

𝔫^{w} = (𝔫 ∩ 𝔫^{w}) ⊕ (𝔫^{−} ∩ 𝔫^{w}),  (4.7)(c)

and therefore that

T^{+} ∩ T^{+,w} = {t ∈ T^{+}|Ad(t) preserves 𝔫 ∩ 𝔫^{w}}.  (4.7)(d)

The representations of T^{+} ∩ T^{+,w} on 𝔫^{−} ∩ 𝔫^{w} and on 𝔫 ∩ 𝔫^{−,w} are contragredient.^{3} It follows that on their common domain T^{+} ∩ T^{+,w}, the two characters 2ρ and 2ρ^{w} differ by the square of a determinant:

2ρ^{w}(t) = 2ρ(t) · [det(Ad(t)|_{𝔫^{−}∩𝔫^{w}})]^{2}  (t ∈ T^{+} ∩ T^{+,w}).  (4.7)(e)

This fact provides a natural isomorphism between the ρ covering of T^{+} ∩ T^{+,w} and the ρ^{w} covering,

(t, z) ↦ (t, z · det(Ad(t)|_{𝔫^{−}∩𝔫^{w}}) (t ∈ T^{+} ∩ T^{+,w}).

In this way we can regard ρ^{w} as a character of the ρ covering (T^{+} ∩ T^{+,w})^{∼}. We can now state the Weyl denominator formula

Δ(t̃) = ∑_{\begin{matrix}w∈W (𝔨,𝔱) \\ t∈T^{+}∩T^{+,w} \end{matrix}}(−1)^{w}ρ^{w}(t̃).  (4.8)

Suppose now that χ is a genuine irreducible representation of T^{+}. Since it’s the end of the day, I will merely assert confidently that there is a natural way to define a genuine representation χ^{w} of (T^{+} ∩ T^{+,w})^{∼}. (The idea is to twist χ by conjugation by an element of K_{0} defining the Weyl group element w. This conjugation will carry T^{+} ∩ T^{+,w} into T^{+}.)

3This is a somewhat subtle point, even in the simplest case 𝔫^{w} = 𝔫^{−}. One way to see it is by introducing appropriate symplectic forms, and using the fact that the determinant of a symplectic linear map is one.

Theorem 4.9 (the Weyl character formula). Suppose T^{+} is a large Cartan subgroup of a compact Lie group K, χ is a genuine dominant irreducible representation of T̃^{+}, and V^{χ} is the corresponding irreducible representation of K. Suppose t is a regular element of T^{+}, and that t̃ is a preimage of t in the ρ covering T̃^{+}. Then the character of V^{χ} at t is

Θ^{χ}(t) = ( ∑_{\begin{matrix}w∈W (𝔨,𝔱) \\ t∈T^{+}∩T^{+,w} \end{matrix}}(−1)^{w}χ^{w}(t)) /Δ(t̃).

This formula could be proved using the Atiyah-Bott-Lefschetz fixed point formula.^{4}

References

[Hump] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

[Vorange] D. Vogan, Unitary Representations of Reductive Lie Groups, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1987.

4But let us not forget the possibility that this tissue of surmise and hypothesis has no basis in reality.