Lectures of Yuri Neretin 2004--2006 . Draft(June 2007)

Preface

Chapter 1. Gaussian integral operators

1.1. Some preliminaries and notations
1.2. Heisenberg group and Weil representation of $Sp(2n, R )$
1.3. Gaussian operators
1.4. Product of Gaussian operators
1.5. Few linear symplectic geometry
1.6. Gaussian operators and Lagrangian subspaces
1.7. Linear relations. Imitation of basic definitions of matrix theory.
1.8. Symplectic category.
1.9. Symplectic category. Details.
1.10. Proof of boundedness. Canonical forms of Gaussian operators.
1.11. Bibliographical remarks.

Chapter 2. Pseudo-Euclidean geometry and groups $ U(p,q)$.


2.1. Geometry of indefinite Hermitian forms.
2.2. Pseudounitary group $ U(p,q)$.
2.3. Cartan matrix balls.
2.4. The space $ U(n)$. Cell decomposition.
2.5. Jordan angles and Hua double ratio.
2.6. Classification of pseudo-unitary and pseudo-self-adjoint operators.
2.7. Indefinite contractions.
2.8. Potapov coordinates.
2.9. Krein--Shmul'yan category. Compressivity.
2.10. Isotropic category. Inverse limits.
2.11. K\"ahler structure on a matrix ball. Some matrix tricks.
2.12. Matrix balls as symmetric spaces.

Chapter 3. Linear symplectic geometry.


3.1. Symplectic groups.
3.2. Group $Sp(2n, R )$ in complex model.
3.3. Matrix balls.
3.4. Conjugacy classes in $Sp(2n, R )$.
3.5. Symplectic contractions.
3.6. Central extensions. Groups $Sp(2n, R )$ and $ U(p,q)$.
3.7. Central extensions.The Krein--Shmul'yan category.
3.8. Geodesic triangles.
3.9. Digression. Central extensions of groups of symplectomorphisms.
3.10. Central extensions. Maslov index.
3.11. Bibliographical remarks to Chapters 2-3.

Chapter 4. Segal--Bargmann transform.


4.1. Fock space.
4.2. Segal-Bargmann transform.
4.3. Spectral analysis of signals.
4.4. Spectral analysis of singularities.
4.5. Symbols of operators
4.6. Perelomov problem.
4.7. Preliminaries on $\theta $-functions.
4.8. Interpolation and Lagrange formula.
4.9. An exotic inversion formula.
4.10. Bibliographical remarks.

Chapter 5. Gaussian operators in Fock spaces.


5.1. Gaussian operators.
5.2. Proof of product formula.
5.3. Boundedness. Spectra of self-adjoint Gaussian operators.
5.4. Bibliographical remarks.

Chapter 6. Gaussian operators. Details.


6.1. Canonical forms and invariants.
6.2. Norms of Gaussian operators.
6.3. Spectra and eigenvectors.
6.4. Quadratic operators and exponents.
6.5. Matching functions and matrix elements of Gaussian operators.
6.6. Bibliographical remarks.

Chapter 7. Spaces of holomorphic functions in matrix balls.


7.1. Reproducing kernels.
7.2. Highest weight representations of $SU(1,1)$
7.3. Berezin scale.
7.4. Hardy spaces.
7.5. Realization of the Weil representation in holomorphic functions on matrix ball.
7.6. Determinantal systems of differential equations.
7.7. The symplectic category and the spaces $\@mathcal H_\alpha $
7.8. Bibliographical remarks.

Chapter 8. Cartier model.


8.1. Zak transform.
8.2.The push-forward of Fourier transform
8.3. Action of $Sp(2n, Z)$.
8.4. Theta-functions and theta-kernels.
8.5. Bibliographical remarks.

Chapter 9. Gaussian operators over finite fields.


9.1. Quadratic forms.
9.2. Fourier transform and Gauss sums.
9.3. Gaussian operators.
9.4. Fast Fourier transform.
9.5. Bibliographic remarks.

Chapter 10. Introduction to classical $p$-adic groups.


10.1. $p$-adic numbers.
10.2. Classification of quadratic forms.
10.3. Lattices.
10.4. Bruhar--Tits trees
10.5. Bruhat--Tits buildings.
10.6. Buildings related to symplectic groups.
10.7. Nazarov category.
10.8. Buildings. General comments

Chapter 11. Weil representation over $p$-adic field


10.1 Gaussian integrals over $p$-adic field.
10.2. Weil representation, $p$-adic case.
10.3. Adeles and the Weil representation of $Sp(2n, Q)$.
10.4. Group $Sp(2n,Q)$ and the real-adelic correspondence.
10.5. Constructions of modular forms.
10.6. Bibliographical remarks.

Addendum A. Classical groups. Notation.



Addendum B. Representations. Several definitions.



Addendum C. Lie groups. Some logical arrows.



Bibliography



Index and notation


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