Introduction to representation theory of classical Lie groups Binyong Sun Academy of Mathematics and Systems Science, Chinese Academy of Sciences The Tsung-Dao Lee Institute 2019.7.19 Binyong Sun Introduction to representation theory of classical Lie groups Contents 1. Classical Lie groups 2. Representations of compact Lie groups 3. Classical invariant theory 4. Infinite dimensional representations 5. Theta correspondences 6. Classical branching law 7. More questions Binyong Sun Introduction to representation theory of classical Lie groups 1.Classical Lie groups Symmetry: Groups and their representations. P. W. Anderson (Nobel prize winner)µ \It is only slightly overstating the case that physics is the study of symmetry". Binyong Sun Introduction to representation theory of classical Lie groups Origin of group theory Babylonian era (2000 B.C.): the quadratic equation x2 + bx + c = 0 has solutions p −b ± b2 − 4c x = : 2 16 century: The cubic and quartic equations can be solved by radicals. AbelõRuffini Theorem(1799!1824): Quintic or higher degree equations can not be solved by radical in general. Binyong Sun Introduction to representation theory of classical Lie groups Question Can a given polynominal equation be solved by radicals? Galois (1830s): establish group theory; answer this question. Evariste´ Galois (1811õ1832) Binyong Sun Introduction to representation theory of classical Lie groups Examples of groupsµ (Z; +; 0) (Bijection(X ; X ); ◦; 1) (GLn(R); ·; 1) Binyong Sun Introduction to representation theory of classical Lie groups Origin of Lie groups Galois : Symetries of polynomial equations −! Groups; Sophus Lie : Symetries of differential equations −! Lie groups. Binyong Sun Introduction to representation theory of classical Lie groups Sophus Lie (1842õ1899) Binyong Sun Introduction to representation theory of classical Lie groups Classical Lie groups Most widely occurring Lie groups in mathematics and physics. Compact classical Lie groups t O(n) = fg 2 GLn(R) j gg = 1ng; t U(n) = fg 2 GLn(C) j gg¯ = 1ng; t Sp(n) = fg 2 GLn(H) j gg¯ = 1ng: Binyong Sun Introduction to representation theory of classical Lie groups Real classical groups General linear group: GLn(R); GLn(C); GLn(H); Real orthogonal groups and so on: ∗ O(p; q); Sp2n(R); U(p; q); O (2n); Sp(p; q): Complex orthogonal groups and complex symplectic groups: On(C); Sp2n(C): Example ( " # " #) t 1p 0 1p 0 O(p; q) := g 2 GLp+q(R) j g g = : 0 −1q 0 −1q Binyong Sun Introduction to representation theory of classical Lie groups Finite dimensional representation theory: 1939 Binyong Sun Introduction to representation theory of classical Lie groups Hermann Weyl, 1885-1955 Binyong Sun Introduction to representation theory of classical Lie groups Two major achievements: Classical invariant theory; Classical branching law. Binyong Sun Introduction to representation theory of classical Lie groups 2.Representations of compact Lie groups Definition Let G be a group. A representation of G is a complex vector space V , together with a linear action G × V ! V ; (g; v) 7! g:v: Notation: G y V . Continuity condition. Linear action: g:(au + bv) = a(g:u) + b(g:v); (gh):u = g:(h:u); 1:u = u: Binyong Sun Introduction to representation theory of classical Lie groups Standard representations: n n 2n GLn(R) y C ; GLn(C) y C ; GLn(H) y C : n n 2n O(n) y C ; U(n) y C ; Sp(n) y C : Binyong Sun Introduction to representation theory of classical Lie groups Analogy Positive integers : Representations prime numbers : Irreducible representations Definition A representation is said to be irreducible if it is nonzero, and has no proper nonzero subrepresentation. + topological condition. Binyong Sun Introduction to representation theory of classical Lie groups G: compact Lie group. Two basic problemsµ Duality problem: Calculate Irr(G) := fIrreducible representation of Gg= ∼ : Spectral decomposition : Given G y V , write V as a sum of irreducible representations. Binyong Sun Introduction to representation theory of classical Lie groups Solution to duality problem: Highest weight theory (Cartan). Example Irr(U(1)) = Z; and more generally, n Irr(U(n)) = f(a1 ≥ a2 ≥ · · · ≥ an) 2 Z g: Binyong Sun Introduction to representation theory of classical Lie groups Elie´ Cartan, 1869-1951 Binyong Sun Introduction to representation theory of classical Lie groups Example of spectral decomposition 2 1 Md U(1) y L (S ) = τk : k2Z Binyong Sun Introduction to representation theory of classical Lie groups 3. Classical invariant theory n n O(n) y R ) O(n) y C[R ]: Proposition n O(n) C[R ] = C[q]; where n X 2 q := xi : i=1 Binyong Sun Introduction to representation theory of classical Lie groups Problem n Decompose O(n) y C[R ]? Hidden symmetry n O(n) × sl2(R) y C[R ]; " # 1 0 n Pn @ h := 7! + i=1 xi ; 0 −1 2 @xi " # 0 1 1 Pn 2 e := 7! − i=1 xi ; 0 0 2 " # 0 0 2 f := 7! 1 Pn @ : 2 i=1 @x2 1 0 i Binyong Sun Introduction to representation theory of classical Lie groups Harmonic polynomials n n O(n) y H[R ] := f' 2 C[R ] j f · ' = 0g 1 M k n = H [R ]; k=0 where k n n H [R ] := f' 2 H[R ] j ' is homogeneous of degree kg: Binyong Sun Introduction to representation theory of classical Lie groups Theorem n O(n) × sl2(R) y C[R ] L1 k n n = k=0 H [R ] ⊗ L(k + 2 ): Binyong Sun Introduction to representation theory of classical Lie groups More generally, n×k n×k O(n) y R ) O(n) y C[R ]: Hidden symmetry n×k O(n) × sp2k (R) y C[R ]; Theorem (Classical invariant theory) n×k M O(n) × sp2k (R) y C[R ] = τ ⊗ θ(τ): τ Similar for other compact classical Lie groups. Binyong Sun Introduction to representation theory of classical Lie groups 4. Infinite dimensional representations Why? Harmonic analysis, quantum mechanics, number theory ··· . Examples in Harmonic analysis. Fourier series: 2 1 U(1) y L (S ): Fourier transform: n 2 n R y L (R ): Automorphic forms: 2 GLn(R) y L (GLn(Z)nGLn(R)): Binyong Sun Introduction to representation theory of classical Lie groups Founders: Isra¨ılMoiseevich Gelfand, 1913-2009 Binyong Sun Introduction to representation theory of classical Lie groups Harish-Chandra, 1923-1983 Binyong Sun Introduction to representation theory of classical Lie groups Unitary representation: Hilbert space + unitary operators. Another example Stone's Theorem fSelfajoint operator on V g = funitary rep. R y V g A 7! (t 7! eitA): Binyong Sun Introduction to representation theory of classical Lie groups Example of irreducible rep.: G = GLn(R), B := fupper triangular matrixg ⊂ G, × χ : B ! C a character. Then 1 G y ff 2 C (G) j f (bg) = χ(b) · f (g); b 2 B; g 2 Gg is a representation which is irreducible for "generic" χ. Binyong Sun Introduction to representation theory of classical Lie groups G: Lie group. Two basic problemsµ Duality problem: Calculate Irr(G) := f\Irreducible rep." of Gg= ∼ ⊃ Irru(G) := fIrreducible unitary rep. of Gg= ∼ : Spectral decomposition : Given G y V , write V as a sum of irreducible representations. Binyong Sun Introduction to representation theory of classical Lie groups Example of duality problem Langlands correspondence × n Irr(GLn(C)) = fcompletely reducible rep. C y C g= ∼ Binyong Sun Introduction to representation theory of classical Lie groups Robert Langlands Binyong Sun Introduction to representation theory of classical Lie groups Examples of spectral decomposition Fourier series: 2 1 M n U(1) y L (S ) = d C · () : n2Z Fourier transform: Z 2 n iξ·() L (R ) = C · e dξ; Rn Binyong Sun Introduction to representation theory of classical Lie groups 5. Theta correspondence Classical invariant theory ! Theta correspondence compact group ! real or p-adic group; polynomial function ! generalized function; finite dim. rep. ! infinite dim. rep.; local symmetry ! global symmetry; H. Weyl ! R. Howe: completion p-adic fields: Q −−−−−−−−! R; Q2; Q3; Q5; Q7; Q11; ··· : Binyong Sun Introduction to representation theory of classical Lie groups Roger Howe Binyong Sun Introduction to representation theory of classical Lie groups Two fundamental conjectures Howe duality conjecture (Howe 1977) ( One-one correspondence; Multiplicity conservation: Conservation relation conjecture of Kudla-Rallis (Kudla-Rallis 1994) Binyong Sun Introduction to representation theory of classical Lie groups Local symmetry: (p+q)×k O(p; q) × sp2k (R) y C[R ]: Global symmetry: (p+q)×k O(p; q) × Spf2k (R) y S(R ): Binyong Sun Introduction to representation theory of classical Lie groups Write (p+q)×k !k : = S(R ) Ω : = fπ 2 Irr(O(p; q)) j HomO(p;q)(!k ; π) 6= 0g; 0 0 0 Ω : = fπ 2 Irr(Spf2k ( )) j Hom (!k ; π ) 6= 0g: R Spf2k (R) Binyong Sun Introduction to representation theory of classical Lie groups Theorem [Howe, JAMS 1989] One-one correspondenceµThe relation 0 Hom (!k ; π⊗π ) 6= 0 O(p;q)×Spf2k (R) b yields a one-one correspondence 0 Irr(O(p; q)) ⊃ Ω $ Ω ⊂ Irr(Spf2k (R)): Multiplicity preservationµfor all (π; π0) 2 Ω × Ω0, 0 Hom (!k ; π⊗π ) = 1: O(p;q)×Spf2k (R) b Binyong Sun Introduction to representation theory of classical Lie groups Theorem (Howe duality conjecture) The same holds for all real or p-adicd classical groups. The real case: Howe, JAMS 1989. p-adic case§p 6= 2: Waldspurger, Proceeding for 60's birthday of Piatetski-Shapiro, 1990. Orthogonal, symplectic, unitary groups (Multiplicity perservation)µLi-Sun-Tian, Invent. Math. (2011), Orthogonal, symplectic, unitary groups: Gan-Takeda, JAMS (2015). The last caseµGan-Sun, Proceeding for 70's birthday of Howe (2017). Binyong Sun Introduction to representation theory of classical Lie groups Summary of theta correspondence Transfer representations of one classical group to another classical group.