Denominator identities and Lie superalgebras Paolo Papi Sapienza - - PowerPoint PPT Presentation

Denominator identities and Lie superalgebras

Paolo Papi

Sapienza Universit` a di Roma

We find an analogue of the Weyl denominator identity for a basic classical Lie superalgebra. joint work with Victor Kac and Pierluigi M¨

• seneder Frajria

Weyl denominator identity

g complex finite-dimensional Lie algebra h Cartan subalgebra ∆ root system of (g, h) W Weyl group of ∆ ∆+ ⊂ ∆ set of positive roots, ρ = 1

2

• α∈∆+ α

Theorem

• α∈∆+

(1 − e−α) =

• w∈W

sgn(w) ew(ρ)−ρ For g = sl(n), Weyl formula is related to the expansion of the Vandermonde determinant.

Basic classical Lie superalgebras

First we want to locate a suitable class of Lie superalgebras to which trying to extend the Weyl formula. These are the Basic classical Lie superalgebras g = g0 ⊕ g1:

• g is simple;
• g0 is a reductive Lie algebra;
• g has a nondegenerate bilinear invariant form which is

symmetric on g0, symplectic on g1 and such that (g0, g1) = 0. h ⊂ g0 Cartan subalgebra g =

α∈∆

gα weight space decomposition ∆ = ∆0 ∪ ∆1 decomposition into even and odd “roots” W Weyl group of ∆0.

Example: gl(m, n).

gl(m, n) = A B C D

• ,

A B C D

• =

A D

• degree 0

+ B C

• degree 1

is a Lie superalgebra w.r.t. [X, Y ] = XY − (−1)deg(X) deg(Y )YX. h = diagonal matrices, h∗ =

m+n

• i=1

Cεi, δj = εm+j, 1 ≤ j ≤ n. (X, Y ) = str(XY ), str( A B C D

• ) = tr(A) − tr(D).

The above bilinear form can be normalized in such a way that (εi, εj) = δij, (δh, δk) = −δhk. ∆0 = ±{εi − εj | 1 ≤ i < j ≤ m} ∪ ±{δi − δj | 1 ≤ i < j ≤ n} ∆1 = ±{εi − δj | 1 ≤ i ≤ m, 1 ≤ j ≤ n}

Why Weyl formula doesn’t work for superalgebras....

Because:

• 1. the restriction of (·, ·) to SpanR∆ is usually indefinite;
• 2. the sets of positive roots are no more W -conjugate.

Corresponding comments:

• 1. One defines the defect d of g as the dimension of a maximal

isotropic subspace of

α∈∆

Rα. It can be shown that d equals the cardinality of a maximal isotropic subset of ∆+ (a subset S ⊂ ∆+ is isotropic if it is formed by linearly independent pairwise orthogonal isotropic roots);

• 2. it won’t be a surprize that formulas do depend on the choice
• f ∆+.

Kac-Wakimoto-Gorelik Theorem

Fix ∆+. Define the Weyl-Kac denominator and superdenominator R+ =

• α∈∆+

0 (1 − e−α)

• α∈∆+

1 (1 + e−α), R− =

• α∈∆+

0 (1 − e−α)

• α∈∆+

1 (1 − e−α)

where ρ = ρ0 − ρ1, ρi = 1

2

• α∈∆+

i α, i = 0, 1.

Theorem

Let ∆+ be any set of positive roots such that a maximal isotropic subset S of ∆+ is contained in the set of simple roots Π corresponding to ∆+. Then eρR± =

• w∈W ♯

sgn±(w)w eρ

• β∈S(1 ± e−β).

where W # is a subgroup of W .

Main result

Definition

A set of positive root ∆+ is called distinguished if the corresponding set of simple roots has exactly one odd root. The following result is proven in the extended abstract, by representation theoretic methods.

Theorem

Let g = g0 ⊕ g1 be a basic classical Lie superalgebra of defect d, where g = A(d − 1, d − 1) is replaced by gl(d, d). Then, for any distinguished set of positive roots, we have

C eρR± = X

w∈W

sgn(w)w eρ (1 ± e−γ1)(1 − e−γ1−γ2) · · · (1 + ±(−1)d+1e−γ1−γ2−...−γd )

where {γ1, . . . , γd} is an explicitly defined maximal isotropic subset of ∆+ and C is an explicit constant.

Comments

• Distinguished sets of positive root exist for any g and have

been classified by Kac.

• For a distinguished ∆+ the hypothesis of KWG Theorem

holds iff d = 1. This is a strong constraint, since, e.g. def gl(m, n) = min{m, n}.

• Distinguished sets of positive root are quite often used when

presenting g as a contragredient Lie superalgebra (roughly speaking: by generators and relations).

• As we shall see, the proof follows quite naturally by Howe

duality (and for type A, in a more elementary way, by Cauchy formulas).

The metaplectic representation

The choice of a set of positive roots ∆+determines a polarization g1 = g+

1 + g− 1 , where g± 1 =

• α∈∆±

1

gα. Hence we can consider the Weyl algebra W (g1) = T(g1)/x ⊗ y − y ⊗ x − (x, y)

• f (g1, ( , )|g1) and construct the left W (g1)-module

M∆+(g1) = W (g1)/W (g1)g+

1 ,

The module M∆+(g1) is also a sp(g1, ( , ))–module with T ∈ sp(g1, ( , )) acting by left multiplication by θ(T) = −1

2

dim g1

i=1

T(xi)xi, where {xi} is any basis of g1 and {xi} is its dual basis w.r.t. ( , ). Since ad(g0) ⊂ sp(g1, ( , )), we

• btain an action of g0 on M∆+(g1).

The metaplectic representation

We have a h-module isomorphism M∆+(g1) ∼ = S(g−

1 ) ⊗ C−ρ1

where S(g−

1 ) is the symmetric algebra of g− 1 . Its h-character is

chM∆+(g1) = e−ρ1

• α∈∆+

1 (1 − e−α).

(1) Upon multiplication by eρ0

α∈∆+

0 (1 − e−α) the r.h.s. of (1)

becomes eρR− So if we are able to determine the g0-character of M∆+(g1) we have a formula for R−.

Example: sl(m, n) and Cauchy formulas

In this case there is essentially a unique distinguished set of positive roots: g+

1 =

B

• . The special feature of this example is

that g−

1 is a g0-module, and the action of g0 on g− 1 is the natural

action of {(A, B) ∈ gl(m + 1) × gl(n + 1) | tr(A) + tr(B) = 0} on (Cm+1)∗ ⊗ Cn+1. Assume m > n. Cauchy formulas give ch(S(g−

1 )) = ch(S((Cm+1)∗ ⊗ Cn+1)) =

• λ

LAm(τ(λ))LAn(λ) where for λ1 ≥ λ2 ≥ . . . ≥ λn+1 we have set λ =

n+1

• i=1

λiδi, τ(λ) = −w0(

n+1

• i=1

λiεi) and w0 is the longest element in the symmetric group W (Am).

Then S(g−

1 ) ⊗ C−ρ1 =

• s1≥s2≥...≥sn+1

LAm×An(−ρ1 − s1γ1 − . . . − sn+1γn+1), γ1 = εm+1 − δ1, γ2 = εm − δ2, . . . . . . , γn+1 = εm−n+1 − δn+1. By the Weyl character formula, we have for s = s1 ≥ s2 ≥ . . . ≥ sn+1

e−ρ1 Q

β∈∆+ 1

(1 − e−β) = X

s

chLAm×An (λs1,...,sn+1 ) = X

s

X

w∈W

sgn(w) ew(λs1,...,sn+1 +ρ0) eρ0 Q

β∈∆+

(1 − e−α) .

Hence eρR− =

• s1≥s2≥...≥sn+1
• w∈W

sgn(w)ew(ρ−s1γ1−...−sn+1γn+1)

• s1,s2,...,sn+1
• w∈W

sgn(w)ew(ρ−s1γ1−s2(γ1+γ2)...−sn+1(γ1+...+γn+1))

General case: Howe duality

For a distinguished set of positive roots, we build up a real form V

• f g1 endowed with a standard symplectic basis {eα, fα}α∈∆+

1 such

that

α∈∆+

1 C(eα ± √−1fα) = g+

1 . It turns out that

sp(V ) ∩ ad(g0) = s1 × s2, si, i = 1, 2, being the Lie algebras of a compact dual pair (G1, G2) in Sp(V ). Distinguished sets of positive roots turn out to correspond in this way to compact dual pairs (G1, G2), with G1 compact: ∆+

B → (O(2m + 1), Sp(2n, R)),

∆+

A → (U(m), U(n)),

∆+

D1 → (O(2m), Sp(2n, R)),

∆+

D2 → (Sp(m), SO∗(2n)).

If λ → τ(λ) is the Theta correspondence, then, as g0-modules ch M∆+(g1) =

• λ

LG1(λ)L(s2)C(τ(λ)).

Updating the main result....

By now we have proven WITH COMBINATORIAL METHODS the following theorem.

Theorem

Let g = g0 ⊕ g1 be a basic classical Lie superalgebra of defect d, where g = A(d − 1, d − 1) is replaced by gl(d, d). Let ∆+ be any set of positive roots and Q+ be the corresponding positive root

• lattice. There is a combinatorial procedure that starting from ∆+

yields a class of maximal isotropic sets in ∆+. Fix any of them, say Sspecial = {γ1, . . . , γd}. Then we have C · eρR± =

• w∈Wg

sgn±(w)w eρ d

i=1(1 ± χ± i e−γi)

for an explicit constant C. Moreover, there exists a choice for Sspecial such that γi ∈ Q+ for any i = 1, . . . , d.

Comments

Undefined notation: ε(η) =

• 1

if η ∈ Z∆0, −1 if η ∈ Z∆ \ Z∆0, γ≤

i = {β ∈ Sspecial, β ≤ γi},

γi =

• β∈γ≤

i

ε(γi − β)β, χ+

i = (−1)|γ≤

i |+1, χ−

i = 1

where β ≤ γ if γ − β is a sum of positive roots or zero. A nice feature of the theorem is that it allows us to recover the Theta correspondence for compact dual pairs starting from the combinatorial formula. This is highly non trivial, but it works!