introduction to representation theory of classical lie groups · 1.classical lie groups...
TRANSCRIPT
![Page 1: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/2.jpg)
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
![Page 3: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/3.jpg)
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
![Page 4: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/6.jpg)
Examples of groupsµ
(Z,+, 0)
(Bijection(X ,X ), , 1)
(GLn(R), ·, 1)
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 7: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/8.jpg)
Sophus Lie (1842õ1899)
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 9: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/10.jpg)
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
![Page 11: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/11.jpg)
Finite dimensional representation theory:
1939
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 12: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/12.jpg)
Hermann Weyl, 1885-1955
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 13: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/13.jpg)
Two major achievements:
Classical invariant theory;
Classical branching law.
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 14: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/16.jpg)
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
![Page 17: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/19.jpg)
Elie Cartan, 1869-1951
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 20: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/20.jpg)
Example of spectral decomposition
U(1) y L2(S1) =⊕
k∈Zτk .
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 21: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/24.jpg)
Theorem
O(n)× sl2(R) y C[Rn]
=⊕∞
k=0Hk [Rn]⊗ L(k + n2 ).
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 25: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/26.jpg)
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
![Page 27: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/27.jpg)
Founders:
Israıl Moiseevich Gelfand, 1913-2009
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 28: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/28.jpg)
Harish-Chandra, 1923-1983
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 29: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/32.jpg)
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
![Page 33: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/33.jpg)
Robert Langlands
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 34: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/34.jpg)
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
![Page 35: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/35.jpg)
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
![Page 36: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/36.jpg)
Roger Howe
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 37: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/37.jpg)
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
![Page 38: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/38.jpg)
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
![Page 39: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/39.jpg)
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
![Page 40: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/40.jpg)
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
![Page 41: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/41.jpg)
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
![Page 42: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/42.jpg)
Summary of theta correspondence
Transfer representations of one classical group to another
classical group.
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 43: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/43.jpg)
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
![Page 44: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/44.jpg)
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
![Page 45: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/45.jpg)
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
![Page 46: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/46.jpg)
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
![Page 47: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/47.jpg)
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
![Page 48: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/48.jpg)
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
![Page 49: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/49.jpg)
6. Classical branching law
Two methods of constructing representationsµ
Induction, restriction.
Restriction↔ Symmetry breaking.
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 50: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/50.jpg)
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
![Page 51: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/51.jpg)
Proof
• Classical invariant theory.
Application
• Basis of irreducible representation.
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 52: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/52.jpg)
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
![Page 53: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/53.jpg)
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
![Page 54: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/54.jpg)
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
![Page 55: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/55.jpg)
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
![Page 56: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/56.jpg)
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
![Page 57: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/57.jpg)
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
![Page 58: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/58.jpg)
L-function: generalization of Riemann zeta function
Langlands program:
Binyong Sun Introduction to representation theory of classical Lie groups
![Page 59: 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](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edcaf36ad6a402d666774d5/html5/thumbnails/59.jpg)
Thank you!
Binyong Sun Introduction to representation theory of classical Lie groups