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 d the structure for � G ( R ) —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 Combinatorics 1 Representations 2 Chao-Ping Dong (HNU) Spin norm September 11, 2018 3 / 38
Outline Combinatorics 1 Representations 2 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 + 2 b ( 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 � A 4 . 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 = λ i ̟ i = [ λ 1 , . . . , λ l ] . i 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. � l s i ( λ ) = λ − λ i j = 1 a ji ̟ 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 = ̟ 1 + · · · + ̟ l = [ 1 , . . . , 1 ] ; and { µ − ρ } is the unique dominant weight to which µ − ρ is conjugate. e.g. {− ρ } = ρ . Thus � 0 � spin = � 2 ρ � . Moreover, � 2 ρ � spin = � 2 ρ � , 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 : P ( µ ) = { µ + n β | n ∈ Z ≥ 0 } , where β is the highest root. e.g. P ( 0 ) consists of 0, β , 2 β, · · · . 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 2 ρ . 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 = ̟ 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 Combinatorics 1 Representations 2 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: � n D := 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 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. 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 . Pandži´ c, 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 . 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... 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, − s λ ) , s ∈ W is an involution, 2 λ is dominant integral. Here µ := { λ + s λ } 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. Chao-Ping Dong (HNU) Spin norm September 11, 2018 24 / 38
Recommend
More recommend
