Spin norm: combinatorics and representations Chao-Ping Dong - - PowerPoint PPT Presentation

Spin norm: combinatorics and representations

Chao-Ping Dong

Institute of Mathematics Hunan University

September 11, 2018

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Overview

This talk aims to introduce the following preprints in 2017.

• J. Ding, C.-P

. Dong, Unitary representations with Dirac cohomology: a finiteness result, arXiv:1702.01876. C.-P . Dong, Unitary representations with Dirac cohomology for complex E6, arXiv:1707.01380. C.-P . Dong, Unitary representations with Dirac cohomology: finiteness in the real case, arXiv:1708.00383. For a real reductive Lie group G(R), we report a finiteness theorem for the structure for G(R)

d

—all the irreducible unitary Harish-Chandra modules (up to equivalence) for G(R) with non-zero Dirac cohomology.

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Outline

1

Combinatorics

2

Representations

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Outline

1

Combinatorics

2

Representations

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A game

The following problem was given at the International Olympiad of Mathematics in 1986. Five integers with positive sum are arranged on a circle. The following game is played. If there is at least one negative number, the player may pick up one of them, add it to its neighbors, and reverse its sign. The game terminates when all the numbers are

• nonnegative. Prove that this game must always terminate.

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Elementary Solution (Demetres Chrisofides)

Take T = (a − c)2 + (b − d)2 + (c − e)2 + (d − a)2 + (e − b)2. After replacing a, b, c by a + b, −b, b + c, we get T ′ = T + 2b(a + b + c + d + e) < T.

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Some examples

The underlying structure: Coxeter group of A4. e.g. consider A2: [−1, −1] → [1, −2] → [−1, 2] → [1, 1]. The Cartan matrix

• 2

−1 −1 2

• Chao-Ping Dong (HNU)

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The A2 picture

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Some examples (continued)

e.g. consider G2: [−1, −1] → [−4, 1] → [4, −3] → [−5, 3] → [5, −2] → [−1, 2] → [1, 1]. The Cartan matrix

• 2

−3 −1 2

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The G2 picture

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The underlying algorithm

Given an arbitrary integral weight λ =

• i

λi̟i = [λ1, . . . , λl]. How to effectively conjugate it to the dominant Weyl chamber? The algorithm: select an arbitrary index i such that λi < 0, then apply the simple reflection si; continue this process when necessary. si(λ) = λ − λi l

j=1 aji̟j. It uses the i-th column of the Cartan

matrix A. Why is the algorithm effective? See Theorem 4.3.1 of A. Björner, F . Brenti, Combinatorics of Coxeter groups, GTM 231, Springer, New York (2005).

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Spin norm (for complex Lie groups)

For any dominant weight µ. The spin norm of µ: µspin = {µ − ρ} + ρ. Here ρ = ̟1 + · · · + ̟l = [1, . . . , 1]; and {µ − ρ} is the unique dominant weight to which µ − ρ is conjugate. e.g. {−ρ} = ρ. Thus 0spin = 2ρ. Moreover, 2ρspin = 2ρ, and ρspin = ρ Note that µspin ≥ µ, and equality holds if and only if µ is

• regular. It becomes subtle and interesting when µ is irregular.

This notion was raised in my 2011 thesis. Origin: Vµ ⊗ Vρ.

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Pencils

The pencil starting with µ: P(µ) = {µ + nβ | n ∈ Z≥0}, where β is the highest root. e.g. P(0) consists of 0, β, 2β, · · · . Reference: D. Vogan, Singular unitary representations, Noncommutative harmonic analysis and Lie groups (Marseille, 1980), 506–535. Motivation: describe the K-types pattern of an infinite-dimensional representation.

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The u-small convex hull (for complex Lie groups)

The u-small convex hull: the convex hull of the W-orbit of 2ρ. Reference: S. Salamanca-Riba, D. Vogan, On the classification of unitary representations of reductive Lie groups, Ann. of Math. 148 (1998), 1067–1133. Motivation: describe a unifying conjecture on the shape of the unitary dual. Pavle’s 2010 Nankai U Lecture: a work joint with Prof. Renard.

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The complex G2 case, where β = ̟2

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Distribution of spin norm along pencils

Theorem

Let g be any finite-dimensional complex simple Lie algebra. The spin norm increases strictly along any pencil once it goes beyond the u-small convex hull. Reference: C.-P . Dong, Spin norm, pencils, and the u-small convex hull, Proc. Amer. Math. Soc. 144 (2016), 999–1013.

Remark

Classical groups: two weeks; Exceptional groups: about two years.

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Outline

1

Combinatorics

2

Representations

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Dirac operator in physics

In 1928, by using matrix algebra, Dirac discovered the later named Dirac operator in his description of the wave function of the spin−1/2 massive particles such as electrons and quarks. Reference: P . Dirac, The quantum theory of the electron, Proc.

• Roy. Soc. London Ser. A 117 (1928), 610–624.

Atiyah’s remark: using Hamilton quaternions H = {±1, ±i, ±j, ±k}, ij = −ji, i2 = −1, we have −∆ = − ∂2 ∂x2 − ∂2 ∂y2 − ∂2 ∂z2 = (i ∂ ∂x + j ∂ ∂y + k ∂ ∂z )2.

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Paul Dirac

Figure 1: Paul Dirac in 1933.

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Dirac operator in Lie theory

In 1972, Parthasarthy introduced the Dirac operator for G and successfully used it to construct most of the discrete series. Reference: R. Parthasarathy, Dirac operators and the discrete series, Ann. of Math. 96 (1972), 1–30. Let {Zi}n

i=1 be an o.n.b. of p0 w.r.t. B. The algebraic Dirac

• perator is defined as:

D :=

n

• i=1

Zi ⊗ Zi ∈ U(g) ⊗ C(p). Note that we have D2 = −(Ωg ⊗ 1 + ρ2) + (Ωk∆ + ρc2).

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Dirac cohomology

Let X be a (g, K)-module. Then D : X ⊗ S → X ⊗ S, and in the 1997 MIT Lie groups seminar, Vogan introduced the Dirac cohomology of X to be HD(X) = Ker D/(Ker D ∩ Im D). Moreover, Vogan conjectured that when HD(X) is nonzero, it should reveal the infinitesimal character of X. This conjecture was verified by Huang and Pandži´ c in 2002. Reference: J.-S. Huang, P . Pandži´ c, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer.

• Math. Soc. 15 (2002), 185–202.

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The classification problem

Problem: classify all the equivalence classes of irreducible unitary representations with non-zero Dirac cohomology. For X unitary, we have HD(X) = Ker D = Ker D2. These representations are extreme ones among the unitary dual in the following sense: they are exactly the ones on which Parthasarathy’s Dirac inequality becomes equality. Cohomological induction is an important way of constructing unitary representations. When the inducing module is one-dimensional, we meet Aq(λ)-modules. Under the admissible condition, J.-S. Huang, Y.-F . Kang, P . Pandži´ c, Dirac cohomology of some Harish-Chandra modules, Transform. Groups. 14 (2009), 163–173.

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Within the good range

The inducing module could be infinite-dimensional. Under the good range condition, C.-P . Dong, J.-S. Huang, Dirac cohomology of cohomologically induced modules for reductive Lie groups, Amer. J. Math. 137 (2015), 37–60. P . Pandži´ c, Dirac cohomology and the bottom layer K-types, Glas.

• Mat. Ser. III 45 (65) (2010), no. 2, 453–460.

What will happen beyond the good range? This point has perplexed us for quite a long time. There could be no unifying formula...

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Complex Lie groups

Let G be a complex connected Lie group, K, H. A powerful reduction: J(λ, −sλ), s ∈ W is an involution, 2λ is dominant integral. Here µ := {λ + sλ} is the LKT. Reference: D. Barbasch, P . Pandži´ c, Dirac cohomology and unipotent representations of complex groups, Noncommutative geometry and global analysis, 1–22, Contemp. Math., 546, Amer.

• Math. Soc., 2011.

Fix λ (say, = ρ/2), and let s varies.

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Complex Lie groups (continued)

Idea: fix an arbitrary involution s, and let λ varies. We call Λ(s) and the corresponding representations J(λ, −sλ) an s-family, where Λ(s) := {λ = [λ1, . . . , λl] | 2λi ∈ P and λ + sλ is integral} . For any involution s ∈ W, put I(s) = {i | s(̟i) = ̟i}. i ∈ I(s) if and only if sαi does not occur in some reduced expression of s, if and only if sαi does not occur in any reduced expression of s. Thus s ∈ sj | j / ∈ I(s). e.g., I(e) = {1, . . . , l}, while I(w0) is empty.

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Complex F4

There are 140 involutions in W(F4). Among them, 103 involutions have the property that I(s) is empty.

• F4

d consists of 10 scattered representations, and 30 strings of

representations.

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Table 1: The scattered part of F d

4

#s λ spin LKT u-small mult 25 [1/2, 1/2, 1/2, 1] [1, 3, 0, 1] Yes 1 38 ρ/2 ρ Yes 1 62 [1, 1, 1/2, 1/2] [0, 0, 1, 4] Yes 1 63 [1/2, 1/2, 1, 1] [7, 1, 0, 0] Yes 1 63 ρ/2 ρ Yes 1 76 [1, 1/2, 1/2, 1] [4, 2, 0, 0] Yes 1 92 [1, 1/2, 1/2, 1/2] [2, 2, 0, 1] Yes 1 122 ρ/2 ρ Yes 1 140 [1, 1, 1/2, 1/2] [0, 0, 0, 4] Yes 1 140 ρ [0, 0, 0, 0] Yes 1

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Table 2: The string part of F d

4 (middle part omitted)

#s λ spin LKT mult 1 [a, b, c, d] LKT 1 2 [1, b, c, d] LKT 1 3 [a, 1, c, d] LKT 1 4 [a, b, 1, d] LKT 1 5 [a, b, c, 1] LKT 1 · · · · · · · · · 1 34 [1, 1, 1/2, d] [3, 0, 0, 2d + 3] 1 34 [1, 1/2, 1/2, d] [1, 2, 0, 2d + 1] 1 47 [1, 1, 1, d] LKT 1 50 [a, 1, 1, 1] LKT 1 50 [a, 1, 1/2, 1/2] [2a + 2, 0, 2, 0] 1 Here a, b, c, d run over the set {1/2, 1, 3/2, 2, . . . }.

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Understanding the string part

C.-P . Dong, On the Dirac cohomology of complex Lie group representations, Transformation Groups 18 (1) (2013), 61–79. Erratum: Transformation Groups 18 (2) (2013), 595–597. Vogan’s encouragement: “...But we are still human, and sometimes we do make mistakes. You feel bad because you are a good mathematician, and that means not accepting errors. Your paper has good mathematics in it..."

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Understanding the string part (continued)

Fix an involution s ∈ W such that I(s) is non-empty. Ps—the θ-stable parabolic subgroup of G corresponding to the simple roots {αi | i / ∈ I(s)}; Ls—the Levi factor. We have that J(λ, −sλ) ∼ = LS(Zλ), where Zλ is the irreducible unitary representation of Ls with Zhelobenko parameters (λ − ρ(us)/2, −s(λ − ρ(us)/2)). The good range condition is met since (λ, −λ), α > 0, ∀α ∈ ∆(us). Reference: D. Vogan, Unitarizability of certain series of representations, Ann. of Math. 120 (1) (1984), 141–187.

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A finiteness result

Theorem (with J. Ding, 2017, arXiv:1702.01876)

The set Gd for a connected complex simple Lie group consists of two parts: a) finitely many scattered modules (the scattered part); and b) finitely many strings of modules (the string part). Moreover, modules in the string part of G are all cohomologically induced from the scattered part of Ld

ss tensored with unitary characters

• f Z(L), and they are all in the good range. Here L runs over the

proper θ-stable Levi subgroups of G, Z(L) is the center of L, and Lss denotes the semisimple factor of L. In particular, there are at most finitely many modules of Gd beyond the good range.

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Some remarks

To classify Gd for G complex, it suffices to consider finitely many candidate representations. Later, we classified Gd for complex E6 (arXiv:1707.01380). The distribution of spin norm along a pencil is very efficient in actual computation. For instance, it reduces the candidate representation in an s-familiy of E6 from 124048 to 3, where s = s4s5s6s5s1s3s2s4s1. Another important tool: atlas, version 1.0, January 2017, see www.liegroups.org for more. Reference: J. Adams, M. van Leeuwen, P . Trapa and D. Vogan, Unitary representations of real reductive groups, preprint, 2012 (arXiv:1212.2192).

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Barbasch-Pandži´ c Conjecture

The following is Conjecture 1.1 of [Barbasch-Pandži´ c, 2010]. Let G be a complex Lie group viewed as a real group, and π be an irreducible unitary representation such that twice the infinitesimal character of π is regular and integral. Then π has nonzero Dirac cohomology if and only if π is cohomologically induced from an essentially unipotent representation with nonzero Dirac

• cohomology. Here by an essentially unipotent representation we

mean a unipotent representation tensored with a unitary character.

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Finiteness in the real case

Theorem (2017, arXiv:1708.00383)

Let G(R) be a real reductive Lie group. For all but finitely many exceptions, any member π in G(R)

d

is cohomologically induced from a member πL(R) in Ld which is in the good range. Here L(R) is a proper θ-stable Levi subgroup of G(R).

• We call the finitely many exceptions the scattered part of

G(R)

d

. The scattered part is the "kernel" of G(R)

d

.

• By [DH-AJM-2015] and cohomological induction in stages, to

classify G(R)

d

for G real reductive, it suffices to consider finitely many candidate representations.

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A few remarks

• We have benefited a lot from the 2017 Atlas workshop held at U of

Utah, July 10–21.

• The powerful reduction due to Barbasch–Pandži´

c is unavailable for real reductive Lie groups yet. We adopted another approach.

• atlas parameter (x, λ, ν), infinitesimal character

1 2(1 + θ)λ + ν ∈ h∗.

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A few conjectures

Conjecture 1. Let G(R) be a real reductive Lie group. Then any spin-lowest K-type of any π in the scattered part of G(R)

d

must be u-small. Conjecture 2. Let G be a connected complex Lie group. The set

• Gd consists exactly of the unitary representations J(λ, −sλ),

where s is an involution, and λ is a weight such that

• 2λ is dominant integral and regular;
• λ + sλ is an integral weight;
• λ − sλ is a non-negative integer combination of simple roots.

Once the Barbasch-Pandži´ c reduction has been worked out for real Lie groups, an analogue of Conj. 2 should be immediate.

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Possible applications

Automorphic forms: Chapter 8 [Huang–Pandži´ c-2006] sharpened the results of [Langlands-1963-AJM] and [Hotta-Parthasarathy-1974-InventMath]. Dirac index polynomial: S. Mehdi, P . Pandži´ c, D. Vogan, Translation principle for Dirac index, Amer. J. Math. 139 (6) (2017), 1465–1491. Other settings.

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Thank you for listening!

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