introduction to representation theory of classical lie groups · 1.classical lie groups...
TRANSCRIPT
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
x =−b ±
√b2 − 4c
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
O(n) = g ∈ GLn(R) | gg t = 1n,U(n) = g ∈ GLn(C) | gg t = 1n,Sp(n) = g ∈ GLn(H) | gg t = 1n.
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
O(p, q) :=
g ∈ GLp+q(R) | g t
[1p 0
0 −1q
]g =
[1p 0
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:
GLn(R) y Cn, GLn(C) y Cn, GLn(H) y C2n.
O(n) y Cn, U(n) y Cn, Sp(n) y C2n.
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 ) := Irreducible representation of G/ ∼ .
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,
Irr(U(n)) = (a1 ≥ a2 ≥ · · · ≥ an) ∈ Zn.
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
U(1) y L2(S1) =⊕
k∈Zτk .
Binyong Sun Introduction to representation theory of classical Lie groups
3. Classical invariant theory
O(n) y Rn ⇒ O(n) y C[Rn].
Proposition
C[Rn]O(n) = C[q],
where
q :=n∑
i=1
x2i .
Binyong Sun Introduction to representation theory of classical Lie groups
Problem
Decompose O(n) y C[Rn]?
Hidden symmetry
O(n)× sl2(R) y C[Rn],
h :=
[1 0
0 −1
]7→ n
2 +∑n
i=1 xi∂∂xi,
e :=
[0 1
0 0
]7→ −1
2
∑ni=1 x
2i ,
f :=
[0 0
1 0
]7→ 1
2
∑ni=1
∂2
∂x2i.
Binyong Sun Introduction to representation theory of classical Lie groups
Harmonic polynomials
O(n) y H[Rn] := ϕ ∈ C[Rn] | f · ϕ = 0
=∞⊕k=0
Hk [Rn],
where
Hk [Rn] := ϕ ∈ H[Rn] | ϕ is homogeneous of degree k.
Binyong Sun Introduction to representation theory of classical Lie groups
Theorem
O(n)× sl2(R) y C[Rn]
=⊕∞
k=0Hk [Rn]⊗ L(k + n2 ).
Binyong Sun Introduction to representation theory of classical Lie groups
More generally,
O(n) y Rn×k ⇒ O(n) y C[Rn×k ].
Hidden symmetry
O(n)× sp2k(R) y C[Rn×k ],
Theorem (Classical invariant theory)
O(n)× sp2k(R) y C[Rn×k ] =⊕τ
τ ⊗ θ(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:
U(1) y L2(S1).
Fourier transform:
Rn y L2(Rn).
Automorphic forms:
GLn(R) y L2(GLn(Z)\GLn(R)).
Binyong Sun Introduction to representation theory of classical Lie groups
Founders:
Israıl Moiseevich 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
Selfajoint operator on V = unitary rep. R y V A 7→ (t 7→ e itA).
Binyong Sun Introduction to representation theory of classical Lie groups
Example of irreducible rep.:
G = GLn(R),
B := upper triangular matrix ⊂ G ,
χ : B → C× a character.
Then
G y f ∈ C∞(G ) | f (bg) = χ(b) · f (g), b ∈ B, g ∈ G
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 ) := “Irreducible rep.” of G/ ∼⊃ Irru(G ) := Irreducible unitary rep. of G/ ∼ .
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
Irr(GLn(C)) = completely reducible rep. C× y Cn/ ∼
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:
U(1) y L2(S1) =⊕
n∈ZC · ( )n.
Fourier transform:
L2(Rn) =
∫Rn
C · e iξ·( ) dξ,
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.
p-adic fields: Q completion−−−−−−−−→ 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:
O(p, q)× sp2k(R) y C[R(p+q)×k ].
Global symmetry:
O(p, q)× Sp2k(R) y S(R(p+q)×k).
Binyong Sun Introduction to representation theory of classical Lie groups
Write
ωk : = S(R(p+q)×k)
Ω : = π ∈ Irr(O(p, q)) | HomO(p,q)(ωk , π) 6= 0,
Ω′ : = π′ ∈ Irr(Sp2k(R)) | HomSp2k (R)
(ωk , π′) 6= 0.
Binyong Sun Introduction to representation theory of classical Lie groups
Theorem [Howe, JAMS 1989]
One-one correspondenceµThe relation
HomO(p,q)×Sp2k (R)
(ωk , π⊗π′) 6= 0
yields a one-one correspondence
Irr(O(p, q)) ⊃ Ω↔ Ω′ ⊂ Irr(Sp2k(R)).
Multiplicity preservationµfor all (π, π′) ∈ Ω× Ω′,
HomO(p,q)×Sp2k (R)
(ωk , π⊗π′) = 1.
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.
Binyong Sun Introduction to representation theory of classical Lie groups
Applications of theta correspondence
Constructions of unitary representations:
• Jian-Shu Li§Invent. Math. (1989)
• Ma-Sun-Zhu§preprint (2017)
Constructions of automorphic representations:
• Howe, Proc. Sympos. Pure Math. (1979)
• Harris-Kudla-Sweet, JAMS (1996)
L-functions:
• Kudla-Rallis, Ann. of Math. (1994)
• Gan-Qiu-Takeda, Invent. Math. (2014)
Binyong Sun Introduction to representation theory of classical Lie groups
Problem. Given π ∈ Irr(O(p, q)),
HomO(p,q)(ωk , π) 6= 0 ? (ωk := S(R(p+q)×k).
Binyong Sun Introduction to representation theory of classical Lie groups
Kulda persistence principle:
HomO(p,q)(ωk , π) 6= 0 ⇒ HomO(p,q)(ωk+1, π) 6= 0.
Howe’s stable range:
k ≥ p + q ⇒ HomO(p,q)(ωk , π) 6= 0.
Binyong Sun Introduction to representation theory of classical Lie groups
First occurrence index:
n(π) := mink | HomO(p,q)(ωk , π) 6= 0.
Example.
n(1) = 0,
n(det) = p + q (Weyl, Rallis, Przebinda).
Binyong Sun Introduction to representation theory of classical Lie groups
Theorem (Kudla-Rallis’s conservation relation conjecture),
Sun-Zhu, JAMS (2015)
n(π) + n(π ⊗ det) = p + q.
The same holds for all real or p-adic classical groups.
Binyong Sun Introduction to representation theory of classical Lie groups
Applications of the conservations relations:
The final proof of Howe duality conjecture
• Gan-Sun, proceeding for 70’s birthday of Howe
(2017)
Explicit calculation of theta correspondence
• Atobe-Gan, Invent. Math. (2017)
Zeros and poles and L-functions
• Yamana, Invent. Math. (2014)
Local Landlands correspondence
• Gan-Ichino, Ann. of Math. (2018).
Binyong Sun Introduction to representation theory of classical Lie groups
6. Classical branching law
Two methods of constructing representationsµ
Induction, restriction.
Restriction↔ Symmetry breaking.
Binyong Sun Introduction to representation theory of classical Lie groups
Theorem (Classical branching law)
Let τµ ∈ Irr(U(n))§then
(τµ)|U(n−1) =⊕ν4µ
τν .
Similar result holds for orthogonal groups.
Binyong Sun Introduction to representation theory of classical Lie groups
Proof
• Classical invariant theory.
Application
• Basis of irreducible representation.
Binyong Sun Introduction to representation theory of classical Lie groups
Uniqueness of branching. For all
τµ ∈ Irr(U(n))§τν ∈ Irr(U(n − 1)),
dim HomU(n−1)(τµ, τν) ≤ 1.
Multiplicity one theorem
Similar results holds for all real or p-adic classical groups.
Conjectured: Bernstein-Rallis, 1980’s
p-adic case: Aizenbud-Gourevitch-Rallis-Schiffmann, Ann. of
Math. (2010)
real case: Sun-Zhu, Ann. of Math. (2012)
Binyong Sun Introduction to representation theory of classical Lie groups
Example (Waldspurger formula and Gross-Zagier formula).
Infinite dimensional representation
GL2(R) y π.
One dimensional representation
GL1(R) y χ.
uniqueness of branching
dim HomGL1(R)(π, χ) = 1.
This is the Rankin-Selberg theory for GL2(R)×GL1(R).
Binyong Sun Introduction to representation theory of classical Lie groups
Jacobi groupµ
GLn(R) n H2n+1(R), Sp2n(R) n H2n+1(R).
Multiplicity one theorem for Jacobi groups)
Similar result holds for real or p-adic Jacobi groups.
Conjectured: Prasad, 1990’s
p-adic case: Sun, Amer. J. of Math. (2012)
real case: Sun-Zhu, Ann. of Math. (2012)
Binyong Sun Introduction to representation theory of classical Lie groups
Example (Tate thesis).
Irreducible representation:
GL1(R) n H3(R) y S(R).
One dimensional representation
GL1(R) y χ.
Uniqueness of homogeneous generalized functions:
dim HomGL1(R)(S(R), χ) = 1.
This is theta correspondence for (GL1(R),GL1(R)).
Binyong Sun Introduction to representation theory of classical Lie groups
Applications of the multiplicity one theorem
Local Gan-Gross-Prasad conjecture
• Hongyu He, Invent. Math. (2017)
Global Gan-Gross-Prasad conjecture
• Wei Zhang, Ann. of Math. (2014)
Proof of Kazhdan-Mazur’s nonvanishing hypothesis
• Sun, JAMS (2017)
Binyong Sun Introduction to representation theory of classical Lie groups
8. More questions
Determine theta correspondence.
Local Gan-Gross-Prasad conjecture
Global Gan-Gross-Prasad cojecture
Unitary dual of classical Lie groups
Algebraic automorphic representations and arithmetic of
L-functions.
Binyong Sun Introduction to representation theory of classical Lie groups
L-function: generalization of Riemann zeta function
Langlands program:
Binyong Sun Introduction to representation theory of classical Lie groups
Thank you!
Binyong Sun Introduction to representation theory of classical Lie groups