Spin norm: combinatorics and representations

Spin norm: combinatorics and representations Chao-Ping Dong Institute of Mathematics Hunan University September 11, 2018 Chao-Ping Dong (HNU) Spin norm September 11, 2018 1 / 38

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 E 6, 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 Ĝ(R)d all the irreducible unitary Harish-Chandra modules (up to equivalence) for G(R) with non-zero Dirac cohomology. Chao-Ping Dong (HNU) Spin norm September 11, 2018 2 / 38

Outline 1 Combinatorics 2 Representations Chao-Ping Dong (HNU) Spin norm September 11, 2018 3 / 38

Outline 1 Combinatorics 2 Representations Chao-Ping Dong (HNU) Spin norm September 11, 2018 4 / 38

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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 5 / 38

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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 6 / 38

Some examples The underlying structure: Coxeter group of Ã4. e.g. consider A 2 : [ 1, 1] [1, 2] [ 1, 2] [1, 1]. The Cartan matrix [ 2 1 1 2 ] Chao-Ping Dong (HNU) Spin norm September 11, 2018 7 / 38

The A 2 picture Chao-Ping Dong (HNU) Spin norm September 11, 2018 8 / 38

Some examples (continued) e.g. consider G 2 : [ 1, 1] [ 4, 1] [4, 3] [ 5, 3] [5, 2] [ 1, 2] [1, 1]. The Cartan matrix [ 2 3 1 2 ] Chao-Ping Dong (HNU) Spin norm September 11, 2018 9 / 38

The G 2 picture Chao-Ping Dong (HNU) Spin norm September 11, 2018 10 / 38

The underlying algorithm Given an arbitrary integral weight λ = i λ i ϖ i = [λ 1,..., λ l ]. How to effectively conjugate it to the dominant Weyl chamber? The algorithm: select an arbitrary index i such that λ i < 0, then apply the simple reflection s i ; continue this process when necessary. s i (λ) = λ λ i l j=1 a jiϖ j. 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). Chao-Ping Dong (HNU) Spin norm September 11, 2018 11 / 38

Spin norm (for complex Lie groups) For any dominant weight µ. The spin norm of µ: µ spin = {µ ρ} + ρ. Here ρ = ϖ 1 + + ϖ l = [1,..., 1]; and {µ ρ} is the unique dominant weight to which µ ρ is conjugate. e.g. { ρ} = ρ. Thus 0 spin = 2ρ. Moreover, 2ρ spin = 2ρ, and ρ spin = ρ Note that µ spin 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: V µ V ρ. Chao-Ping Dong (HNU) Spin norm September 11, 2018 12 / 38

Pencils The pencil starting with µ: where β is the highest root. P(µ) = {µ + nβ n Z 0 }, 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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 13 / 38

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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 14 / 38

The complex G 2 case, where β = ϖ 2 Chao-Ping Dong (HNU) Spin norm September 11, 2018 15 / 38

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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 16 / 38

Outline 1 Combinatorics 2 Representations Chao-Ping Dong (HNU) Spin norm September 11, 2018 17 / 38

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, i 2 = 1, we have = 2 x 2 2 y 2 2 z 2 = (i x + j y + k z )2. Chao-Ping Dong (HNU) Spin norm September 11, 2018 18 / 38

Paul Dirac Figure 1: Paul Dirac in 1933. Chao-Ping Dong (HNU) Spin norm September 11, 2018 19 / 38

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 {Z i } n i=1 be an o.n.b. of p 0 w.r.t. B. The algebraic Dirac operator is defined as: D := n Z i Z i U(g) C(p). i=1 Note that we have D 2 = (Ω g 1 + ρ 2 ) + (Ω k + ρ c 2 ). Chao-Ping Dong (HNU) Spin norm September 11, 2018 20 / 38

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 H D (X) = Ker D/(Ker D Im D). Moreover, Vogan conjectured that when H D (X) is nonzero, it should reveal the infinitesimal character of X. This conjecture was verified by Huang and Pandžić in 2002. Reference: J.-S. Huang, P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185 202. Chao-Ping Dong (HNU) Spin norm September 11, 2018 21 / 38

The classification problem Problem: classify all the equivalence classes of irreducible unitary representations with non-zero Dirac cohomology. For X unitary, we have H D (X) = Ker D = Ker D 2. 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 A q (λ)-modules. Under the admissible condition, J.-S. Huang, Y.-F. Kang, P. Pandžić, Dirac cohomology of some Harish-Chandra modules, Transform. Groups. 14 (2009), 163 173. Chao-Ping Dong (HNU) Spin norm September 11, 2018 22 / 38

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. Pandžić, 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... Chao-Ping Dong (HNU) Spin norm September 11, 2018 23 / 38

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. Pandžić, 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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 24 / 38

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 = [λ 1,..., λ l ] 2λ i P and λ + sλ is integral}. For any involution s W, put I(s) = {i s(ϖ i ) = ϖ i }. i I(s) if and only if s αi does not occur in some reduced expression of s, if and only if s αi does not occur in any reduced expression of s. Thus s s j j / I(s). e.g., I(e) = {1,..., l}, while I(w 0 ) is empty. Chao-Ping Dong (HNU) Spin norm September 11, 2018 25 / 38

Complex F 4 There are 140 involutions in W (F 4 ). Among them, 103 involutions have the property that I(s) is empty. F 4 d consists of 10 scattered representations, and 30 strings of representations. Chao-Ping Dong (HNU) Spin norm September 11, 2018 26 / 38

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 Chao-Ping Dong (HNU) Spin norm September 11, 2018 27 / 38

Table 2: The string part of F 4 d (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,... }. Chao-Ping Dong (HNU) Spin norm September 11, 2018 28 / 38

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..." Chao-Ping Dong (HNU) Spin norm September 11, 2018 29 / 38

Understanding the string part (continued) Fix an involution s W such that I(s) is non-empty. P s the θ-stable parabolic subgroup of G corresponding to the simple roots {α i i / I(s)}; L s the Levi factor. We have that J(λ, sλ) = L S (Z λ ), where Z λ is the irreducible unitary representation of L s with Zhelobenko parameters (λ ρ(u s )/2, s(λ ρ(u s )/2)). The good range condition is met since (λ, λ), α > 0, α (u s ). Reference: D. Vogan, Unitarizability of certain series of representations, Ann. of Math. 120 (1) (1984), 141 187. Chao-Ping Dong (HNU) Spin norm September 11, 2018 30 / 38

A finiteness result Theorem (with J. Ding, 2017, arxiv:1702.01876) The set Ĝd 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 L d ss tensored with unitary characters of 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 L ss denotes the semisimple factor of L. In particular, there are at most finitely many modules of Ĝd beyond the good range. Chao-Ping Dong (HNU) Spin norm September 11, 2018 31 / 38

Some remarks To classify Ĝd for G complex, it suffices to consider finitely many candidate representations. Later, we classified Ĝd for complex E 6 (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 E 6 from 124048 to 3, where s = s 4 s 5 s 6 s 5 s 1 s 3 s 2 s 4 s 1. 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). Chao-Ping Dong (HNU) Spin norm September 11, 2018 32 / 38

Barbasch-Pandžić Conjecture The following is Conjecture 1.1 of [Barbasch-Pandžić, 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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 33 / 38

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 Ĝ(R)d is cohomologically induced from a member π L(R) in L d 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 Ĝ(R)d. The scattered part is the "kernel" of Ĝ(R)d. By [DH-AJM-2015] and cohomological induction in stages, to classify Ĝ(R)d for G real reductive, it suffices to consider finitely many candidate representations. Chao-Ping Dong (HNU) Spin norm September 11, 2018 34 / 38

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 Pandžić 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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 35 / 38

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 Ĝ(R)d must be u-small. Conjecture 2. Let G be a connected complex Lie group. The set Ĝ d 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-Pandžić reduction has been worked out for real Lie groups, an analogue of Conj. 2 should be immediate. Chao-Ping Dong (HNU) Spin norm September 11, 2018 36 / 38

Possible applications Automorphic forms: Chapter 8 [Huang Pandžić-2006] sharpened the results of [Langlands-1963-AJM] and [Hotta-Parthasarathy-1974-InventMath]. Dirac index polynomial: S. Mehdi, P. Pandžić, D. Vogan, Translation principle for Dirac index, Amer. J. Math. 139 (6) (2017), 1465 1491. Other settings. Chao-Ping Dong (HNU) Spin norm September 11, 2018 37 / 38

Thank you for listening! Chao-Ping Dong (HNU) Spin norm September 11, 2018 38 / 38