arXiv:1703.07238v1 [math.RT] 21 Mar 2017

On the Weil representation of
infinite-dimensional symplectic group over a
finite field
Yu.A.Neretin1
We extend the Weil representation of infinite-dimensional symplectic group to a
representation a certain category of linear relations.

1

The statement of the paper

1.1. The spaces V2µ . Denote by C× be the multiplicative group of complex numbers. Denote by F a finite field of characteristic p > 2, by F× its
multiplicative group. Fix a nontrivial character Exp(·) of the additive group of
F,
Exp(a + b) = Exp(a) Exp(b),
Exp(·) ∈ C× .
Values of Exp(·) are e2πik/p .
For n = 0, 1, . . . , we denote by Vnd an n-dimensional linear space over F, to
be definite we assume that Vnd is the coordinate space Fn . Denote by Vnc the
second copy of this space.
d
Next, we define the space V∞
as a direct sum of a countable number of F
c
and the space V∞ as a direct product of a countable number of F. We equip the
first space by the discrete topology, the second space by the product-topology
(’d’ is an abreviation of ’discrete’ and ’c’ of ’compact).
Below subscripts µ, ν, κ denote 0, 1, . . . , ∞.
A space Vµc is the Pontryagin dual of Vµd , the pairing is given by
[x, y] :=

µ
X

xj yj .

j=1

(actually, the sum is finite). For a subspace L ⊂ Vµd we denote by L◦ the
anihilator of L in Vµc and vice versa.
Denote by V2µ the space
V2µ := Vµd ⊕ Vµc
equipped with the following skew-symmetric bilinear form (symplectic form):

1 Supported

(x, y), (x′ , y ′ ) =

µ
X
j=1

by the grant FWF, Project P28421.

1

(xj yj′ − yj x′j ).

The space V2∞ is a locally compact Abelian group and we can apply the Pontryagin duality.
For a subspace R ⊂ V2µ we denote by R♦ the orthocomplement with respect
to {·, ·}. By the Pontryagin duality (see [7], Proposition 38), for closed subspaces
X, Y ⊂ V2∞ the conditions X = Y ♦ and Y = X ♦ are equivalent (in other words
for a closed subspace X we have X ♦♦ = X).
Denote by Sp(V2µ ) the group of invertible continuous operators in V2µ preserving the form {·, ·}.
1.2. The Weil representation of Sp(V2∞ ). Consider the space ℓ2 (Vµd )
on the set Vµd . For any v = (v d , v c ) ∈ V2µ we define a unitary operator a(v) in
ℓ2 (Vµd ) given by
a(v) f (x) = f (x + v d ) Exp
We have
a(v)a(w) = Exp

hX

1
2 {v, w}

i
xj vjc + 21 vjd vjc .



a(v + w).

Thus we get a projective unitary representation of the additive group V2µ or a
linear representation of the Heisenberg group, the latter group is V2µ ⊕ F with
a product

(v; s) ⋆ (w; t) = v + w; s + t + 21 {v, w} .
Proposition 1.1 For any g ∈ Sp(V2∞ ) there exists a unique up to a scalar
d
factor unitary operator W (g) in ℓ2 (V∞
) such that
a(vg) = W (g)−1 a(v)W (g)

(1.1)

and
W (g1 )W (g2 ) = c(g1 , g2 )W (g1 g2 ),

where c(g1 , g2 ) ∈ C, |c(g1 , g2 )| = 1.

d
This statement is inside the original construction of A. Weil (the group V∞
is locally compact), in Section 2 we present a proof (this proof is used in Section
3).
Remark. The author does not know is the Weil representation of Sp(V2∞ )
projective or linear, for a finite symplectic group Sp(2n, F) it is linear, see, e.g.,
[4].
⊠

1.3. Linear relations. Let X, Y be linear spaces over F. A linear relation
T : X ⇒ Y is a linear subspace in X ⊕ Y . Let T : X ⇒ Y , S : Y ⇒ Z be linear
relations. Their product ST is defined as the set of all (x, z) ∈ X ⊕ Z such that
there exists y ∈ Y satisfying (x, y) ∈ T , (y, z) ∈ S.
Remark. Let A : X → Y be a linear operator. Then its graph graph(A)
is a linear relation X ⇒ Y . If A : X → Y , Y → Z are linear operators, then
product of their graphs
graph(A) : X ⇒ Y,

graph(B) : Y ⇒ Z
2

is the linear relation graph(BA) : X ⇒ Z.

⊠

For a linear relation T : X ⇒ Y we define:
• the kernel ker T is the intersection T ∩ (X ⊕ 0);
• the domain dom T is the image of the projection of T to X ⊕ 0;
• the image im T is the image of the projection of T to 0 ⊕ Y ;
• the indefiniteness indef T is the intersection T ∩ (0 ⊕ Y ).
We define the pseudo-inverse relation T  : Y ⇒ X as the image of T under
the transposition of summands in X ⊕ Y .
For a linear relation T : X ⇒ Y and a subspace U ⊂ X we define a subspace
T U ⊂ Y consisting of all y ∈ Y such that there exists u ∈ U satisfying (u, y) ∈ T .
Remark. For a linear operator A : X → Y we have
ker graph(A) = ker A,

dom graph(A) = X,

im graphA = im(A),

indef graph(A) = 0.

If A is invertible, then graph(A) = graph(A−1 ). For a subspace U ⊂ X we
have graph(A)U = AU .
⊠
1.4. Perfect Lagrangian linear relations. We say that a subspace
Z ⊂ V2∞ is codiscrete if it is closed and the quotient V2∞ /Z is discrete.
Let µ, ν = 0, 1, . . . , ∞. Define a skew-symmetric bilinear form in V2µ ⊕ V2ν
by



(v, w), (v ′ , w′ ) V2µ ⊕V2ν = v, v ′ V2µ − w, w′ V2ν .

We say that a linear relation T : V2µ ⇒ V2ν is perfect Lagrangian if it
satisfies the following conditions:
1) subspace T is maximal isotropic in V2µ ⊕ V2ν (in particular, T is closed);
2) ker T and indef T are compact;

3) im T , dom T are codiscrete.
Remark. Let A ∈ Sp(V2µ ). Then graph(A) is a perfect Lagrangian linear
relation V2µ ⇒ V2µ .
⊠
Lemma 1.2 For any perfect Lagrangian linear relation T ,
(dom T )♦ = ker T,

(ker T )♦ = dom T,

(im T )♦ = indef T,

(indef T )♦ = im T.

Lemma 1.3 Let T : V2µ → V2ν be a perfect Lagrangian linear relation and
ker T = 0, indef T = 0. Then µ = ν and T is a graph of an element of Sp(V2µ ).
Theorem 1.4 If linear relations T : V2µ ⇒ V2ν , S : V2ν ⇒ V2κ are perfect
Lagrangian, then ST : V2µ ⇒ V2κ is perfect Lagrangian.

3

We define a category L, whose objects are the spaces V2µ and morphisms
are perfect Lagrangian relations. It is more convenient to think that objects of
the category are linear spaces, which are equivalent to the spaces V2µ as linear
spaces equipped with topology and skew-symmetric bilinear form.
1.5. Extension of the Weil representation.
Theorem 1.5 a) Let T : V2µ ⇒ V2ν be a perfect Lagrangian linear relation.
Then were exists a unique up to a scalar factor non-zero bounded operator
W (T ) : ℓ2 (Vµd ) → ℓ2 (Vνd ) such that
a(w)W (T ) = W (T )a(v),

for any (v, w) ∈ T .

(1.2)

b) For any perfect Lagrangian linear relations T : V2µ ⇒ V2ν , T : V2ν ⇒
V2κ ,
W (S) W (T ) = c(T, S) W (ST ),
where c(T, S) ∈ C× .
c) For any perfect Lagrangian T : V2µ ⇒ V2ν ,
W (T )∗ = W (T  ).
1.6. Gaussian operators. Let H ⊂ Vµd ⊕ Vνd be a subspace such that
projection operators H → Vνd ⊕ 0 and H → 0 ⊕ Vµd have finite dimensional
kernels. Let Q be a quadratic form on H. We define a Gaussian operator
G(H; Q) : ℓ2 (Vµd ) → ℓ2 (Vνd ) by the formula
G(H; Q)f (x) =

X

y: (y,x)∈H


Exp Q(x, y) f (y).

(1.3)

Theorem 1.6 Any operator W (T ) : ℓ2 (Vµd ) → ℓ2 (Vνd ) is a Gaussian operator
up to a scalar factor. Any Gaussian operator has such a form.
1.7. Bibliographic remarks. 1) In 1951, K. O. Friedrichs [3] noticed
that the Stone–von Neumann theorem implies the existence of a representation
of a real symplectic group Sp(2n, R). The finite-dimensional groups Sp(2n, R)
were not interesting to him and he formulated a conjecture about an infinitedimensional symplectic group. The representation of Sp(2n, R) was explicitly
described by I. Segal in 1959, [16]. A. Weil extended this construction to groups
over p-adic and finite fields.
2) For infinite-dimensional symplectic group the ’Weil representation’ (which
was conjectured by Friedrichs) was constructed by D. Shale [17] and F. A. Berezin
[1], [2]. The p-adic case was examined by M. L. Nazarov [9], [8] and E. I. Zelenov
[19].
3) The third part of the topic was a categorization of the Weil representation,
it had a long pre-history (see [6]), for real and p-adic infinite-dimensional groups
it was obtained by M.L.Nazarov, G.I.Olshanski, and the author in [9], 1989 (see
more detailed expositions of different topics in [10], [8], [14], [11]).
4

For Sp(2n, F) the corresponding representation of the semigroup of linear
relations was done in the preprint of R.Howe (1976), which was not published
as a paper. A detailed exposition of this representation is contained in2 [12],
Section 9.4.
4) Now there exists well-developed representation theory of infinite-dimensional
classical groups and infinite symmetric groups. Numerous efforts to develop a
parallel representation theory of infinite-dimensional classical groups over finite
fields met some obstacles, which apparently, were firstly observed by Olshanski
in [14], 1991. For a collection of references on different approaches to this topic,
see [13]). In [13] there was proposed a version of GL(∞) over finite fields as a
group of continuous operators in V2∞ , it seems that this group is an interesting
and hand object from the point of view of representation theory. In particular,
there is a natural GL(V2∞ )-invariant measure on an infinite-dimensional Grassmannian (and also on flag spaces) over F and explicit decomposition of L2 on
this Grassmannian. Clearly, this topic is related to the group Sp(V2∞ ) and this
requires an examination of the Weil representation.
1.8. Acknowledges. This note is a kind of addition to the work by
Nazarov, Neretin, and Olshanski [9], [10], [8], [14]. I am grateful to M.L.Nazarov
and G.I.Olshanski. I also thank R.Howe who recently sent me preprint [5].
1.9. Structure of the paper. In Section 2 we prove Proposition 1.1.
The proof follows I.Segal arguments [16], see also [12], Section 1.2. The only
additional difficulty is Proposition 2.2 about generators of the group Sp(V2∞ ).
Statements on linear relations are established in Section 3. A proof of the
theorem about product of perfect Lagrangian relations is based on all the remaining statements, this makes a structure of the section slightly sophisticated.
Lemmas 1.2 and 1.3 are proved in Subs. 3.1–3.3, Theorem 1.5.a in Subs. 3.4.
Properties of Gaussian operators (Theorem 1.6) are established in Subs. 3.6–
3.8. Theorem 1.4 is proved in Subs. 3.9. The remaining part of Theorem 1.5 is
proved in Subs. 3.10.

The Weil representation of the group Sp(V2∞).

2

Here we prove Proposition 1.1.
2.1. Preliminary remarks on linear transformations in V2∞ . Denote
by GL(V2∞ ) the group of all continuous invertible linear transformations v 7→ vg
of V2∞ . Let


a b
g=
∈ GL(V2∞ ).
(2.1)
c d
Then the block c contains only finite number of nonzero elements; each row of
a contains only a finite number of nonzero elements, and each comumn of d
contains a finite number of nonzero elements.
We need some statements from [10], Sect. 2.
2I

refered to [9] being sure that case of finite fields was considered in that paper.

5

A. Let g : V2∞ → V2∞ be a continuous surjective linear transformation in
V2∞ . Then the inverse transformation is continuous.
d
c
B. We say that a continuous linear transformation P in V∞
or V∞
is Fredholm
d
if dim ker P < ∞, im P is closed (this condition is nontrivial only for V∞
), and
codim im P < ∞. The index of a Fredholm operator P is
ind P := dim ker P − codim im P.
The following statements hold:
c
d
1. An operator P in V∞
is Fredholm iff the dual operator P t in V∞
is
Fredholm.
2. Let P be Fredholm and H have a finite rank. Then P + H is Fredholm
and
ind(P + H) = ind P.
c
d
3. If P , Q are Fredholm linear transformations of V∞
or V∞
, then

ind P Q = ind P + ind Q.


a
C. For any matrix g =
c
Fredholm and ind d = − ind a.


b
∈ GL(V2∞ ) (see (2.1)) the blocks a, d are
d

2.2. Fredholm indices of blocks of g ∈ Sp(V2µ ). Elements g ∈ Sp(V2µ ) ⊂
GL(V2µ ) satisfy obvious additional conditions, below we refer to
ct a = act ;

(2.2)

t

(2.3)

t

ad − bc = 1.

a
Lemma 2.1 Let g =
c

b
d



∈ Sp(V2µ ). Then ind a = ind d = 0.

Proof. For g ∈ Sp(V2µ ), we have
ind(adt ) = ind(a) + ind(dt ) = ind(a) − ind(d) = 2 ind(a).
On the other hand, bct has finite rank. By (2.3),
ind(adt ) = ind(1 + bct ) = ind(1) = 0.
Thus, ind(a) = 0.



2.3. Generators of the group Sp(V2∞ ). We define
groups in Sp(V2∞ ):

a
— the subgroup H consists of matrices of the form
0

1
— the subgroup N+ consists of matrices of the form
0
6

the following sub
;
at−1

b
, where b = bt ;
1
0

— the subgroup N− consists of matrices of the form


1
c


0
, where c = ct .
1

Denote by Jk the following block matrix of the size ((k − 1) + 1 + ∞) + ((k −
1) + 1 + ∞):


1 0 0 0 0 0
0 0 0 0 1 0 


0 0 1 0 0 0 

.
Jk = 

0 0 0 1 0 0 
0 −1 0 0 0 0
0 0 0 0 0 1
Proposition 2.2 The group Sp(V2∞ ) is generated by the subgroup H, N+ , and
elements Jk .

 2.4.
 Generators of Sp(V2∞ ). Proof of Proposition 2.2. Let s =
1 0
∈ N− . Since Q contains only a finite number of nonzero elements, we
c 1
have
−1
J1 . . . Jm sJm
. . . J1−1 ∈ N+
for sufficiently large m. Thus, N− is contained in the subgroup generated by
H, N+ , and Jk .
Let


a b
g=
∈ Sp(V2∞ ).
c d
We will multiply it by elements of groups H, N+ , N− and elements Jk in an
appropiate way. As a result we will come to the unit matrix.
Recall that a is Fredholm of index 0. Therefore there are invertible operators
d
K,
 L in
 V∞ such that KaL is a block (l + ∞) × (l + ∞)-matrix of the form
0 0
. This follows from [13], Lemma 2.7. We pass to a new matrix g ′ given
0 1
by


0
0 b′11 b′12




0
K
0
a b
L
0
1 b′21 b′22 

g ′ :=
=
t−1
t−1
′
c11 c′12 d′11 d′11 
0 K
c d
0 L
c′21 c′22 d′21 d′22
(the size of the right hand side is l + ∞ + l + ∞). The condition (2.2) implies
c′12 = 0, c′21 = 0, (c′22 )t = c′22 . In particular the following matrix


1 0
0
0
0 1
0
0

s := 
0 0
1
0
0 0 −c′22 1

7

is contained in N− . The element sg ′ has the form


0 0 b′′11 b′′12
 0 1 b′′21 b′′22 
′′

sg ′ = 
c′′11 0 d′′11 d′′11  =: g .
0 0 d′′21 d′′22

The matrix c′′11 is nondegenerate (otherwise, the whole matrix sg ′ is degenerate).
We take an element
J1 . . . Jl g ′′ =: g ′′′
and get a matrix of the form

g ′′′

b′′′
11
b′′′
21
d′′′
11
d′′′
21


b′′′
12

b′′′
22 
′′′  .
d12
d′′′
12

 ′′
−c11
=
0

−1
0
.
1



−c′′11
 0
=
 0
0

0
1
0
0

Keeping in the mind (2.3) we observe
 ′′′
d11
d′′′
21
Element

is contained in N+ .



−c′′′
11
 0

 0
0

d′′′
12
d′′′
12
0
1
0
0



0
0
−1
−(c′′′
11 )
0

−1
0
0
 g ′′′
0
1



2.5. Construction of the Weil representation of the symplectic
groups Sp(V2∞ ). A remaining part of a proof of Proposition 1.1 is based on
standard arguments (see [12], Sect. 1.2), which were proposed by I. Segal).
d
) is irreStep 1. The representation a(v) of the Heisenberg group in ℓ2 (V∞
ducible. Indeed, let us show that there are no nontrivial intertwining operators.
d
Any bounded operator Q in ℓ2 (V∞
) commuting with all operators

X
a(0, v c )f (x) = Exp
vjc xj f (x)
is an operator of multiplication by a function. Since Q commutes also with
shifts
a(v d , 0)f (x) = f (x + v d ),

we get that Q is a multiplication by a constant.
Step 2. An operator W (g) is defined up to a scalar factor (if it exists).
Indeed, the map v 7→ vg is an automorphism of the Heisenberg group. Therefore,
the formula v 7→ a(vg) determines a unitary representation of the Heisenberg

8

group. The operator W (g) intertwines unitary representations a(v) and a(vg).
By the Schur lemma, W (g) is unique up to a scalar factor.
Step 3. If W (g1 ), W (g2 ) exist, then
W (g1 )W (g2 ) = λ · W (g1 g2 ).
Indeed,

−1
W (g1 )W (g2 )
a(v)W (g1 )W (g2 ) = W (g2 )−1 W (g1 )−1 a(v)W (g1 ) W (g2 ) =
= W (g2 )−1 a(vg1 )W (g2 ) = a(vg1 g2 ).

Step 4. It remains to write operators corresponding to generators of the
group Sp(V2µ ).
— Operators corresponding to elements of the subgroup H are


a
0
W
f (x) = f (xa);
0 at−1
— Operators corresponding to elements of N+ are


X

1 b
W
f (x) = Exp 21
bkl xk xl f (x).
0 1

(2.4)

— An operator corresponding to Jk is the Fourier transform with respect to
the coordinate xk .

3

The Weil representation of the category of Lagrangian relations

3.1. On a canonical form of a compact isotropic submodule.
Lemma 3.1 For any compact isotropic subspace M ⊂ V2∞ there is an element
c
g ∈ Sp(V2∞ ) such that M g ⊂ V∞
. Moreover, we can choose g is such a way
c
that M g ⊂ V∞ be a subspace given by a system of equations of the type yα = 0,
where α ranges in a subset A ⊂ N.
c
Proof. Consider the projection map π : M → V∞
. Its fibers are compact and discrete, therefore they are finite. In particular, π −1 (0) is a finited
dimensional subspace in V∞
. We can choose an element h of the subgroup
−1
d
H ⊂ Sp(V2∞ ) such that (π (0))h ⊂ V∞
is the subspace consisting of vectors
(x1 , . . . , xk , 0, . . . ). Since M h is isotropic, π(M h is contained in the subspace
c
y1 = · · · = yk = 0. Therefore M hJ1 . . . Jk ⊂ V∞
.
c
Next, we wish to describe closed subspaces in V∞
modulo linear transforc
mations of V∞ (they are induced by elements of the subgroup H). The Pontryagin duality determines a one-to-one correspondence between sets of closed
d
c
(see [7], Theorem 27); the both groups
subgroups in Abelian groups V∞
and V∞

9

are equipped with an Abelian group of automorphisms x 7→ λx, where λ ∈ F× .
It is easy to see that the correspondence send invariant subgroups (subspaces)
to invariant subgroups. Therefore the question is reduced to a description of
d
subspaces in V∞
modulo linear transformations, the latter problem is trivial. 
Corollary 3.2 Let L be a compact isotropic submodule of V2∞ . Then the space
L♦ /L is isomorphic as a symplectic space to V2n or V2∞ .
Lemma 3.3 For any codiscrete coisotropic submodule L ⊂ V2∞ there exists
c
g ∈ Sp(V2∞ ) such that Lg ⊃ V∞
.
Proof. We reduce the compact isotropic module L♦ to the canonical form
c
(i.e., L♦ is a coordinate subspace in V∞
).

3.2. Proof of Lemma 1.2. Consider a perfect Lagrangian relation T .
Obviously, for v ∈ ker T , Z ∈ dom T , we have {v, z} = 0, i.e., (dom T )♦ ⊃
ker T . Next, (dom T )♦ = ker T ; otherwise, we take an isotropic linear relation
T + (dom T )♦ ) T .

3.3. Proof of Lemma 1.3. Only the case µ = ν = ∞ requires a proof. Let
T : V2∞ ⇒ V2∞ be a perfect Lagrangian linear relation, ker T = 0, indef T = 0.
According Lemma 1.2, we have dom T = V2∞ , im T = V2∞ .
Lemma 3.4 The subspace T is isomorphic to V2∞ as a linear space with a
topology.
Proof of Lemma 1.3. We apply statement A of Subsect. 2.1. The
projections
πT : T → V2∞ ⊕ 0,
πT′ : T → 0 ⊕ V2∞
are continuous, therefore the inverse map πT−1 is continuous. Hence the map
πT′ ◦ πT−1 is continuous, this is the the linear transformation whose graph is T .

Proof of Lemma 3.4. Represent V2∞ as a union of a chain of compact
subspaces
c
W0 = V∞
⊂ W1 ⊂ W2 ⊂ . . .
Consider the projection map π : V2∞ ⊕ V2∞ → V2∞ ⊕ 0, denote by πT its
restriction to T . Then


c
c
= ∪∞
πT−1 V∞
j=1 T ∩ (V∞ ⊕ Wj ) .

Therefore



c
c
V∞
= ∪∞
j=1 πT T ∩ (V∞ ⊕ Wj ) .

In the right hand side we have a union of an increasing sequence of compact
sets, the left hand side is compact. Therefore for some k,


c
c
⊕ Wk ) .
V∞
= πT T ∩ (V∞
10

c
The set T ∩ (V∞
⊕ Wk ) is compact and πT is continuous, therefore the inverse
−1
c
c
map πT : V∞ → T ∩ (V∞
⊕ Wk ) is continuous.
d
Next, denote by el the standard basis in V∞
. Then
∞

 M
c
T ≃ T ∩ (V∞
⊕ Wk ) ⊕
F · πT−1 el .
l=1

Thus T is isomorphic V2∞ ⊕ 0.



3.4. Proof of Theorem 1.5.a. Existence of operators W (T ).
Lemma 3.5 Let the claim of Theorem 1.5.a hold for a linear relation T : V2µ ⇒
V2ν . Then the same statement holds for any linear relation gT h, where g ∈
Sp(V2ν ), h ∈ Sp(V2µ ).
Proof. Obvious.



We start a proof of the theorem. Keeping in the mind Lemma 3.1 we can
assume
ker T ⊂ Vµc ,
indef T ⊂ Vνc .
Let w ∈ indef T . Then
a(w) W (T ) = W (T )a(0) = W (T ).
The operator a(w) is an operator of multiplication by a function,

X
xj wjc f (x)
a(w)f (x) = Exp


P
Therefore any function ψ ∈ im W (T ) vanishes on the set Exp
xj wjc 6= 1.
In other words, all elements of ψ ∈ im W (T ) are supported by the subspace
(indef T )◦ ⊂ Vνd .
Let v ∈ ker T . Then
W (T ) = W (T )a(v).
The operator a(v) is a multiplication by function taking finite number of values
λl = exp(2πli/p). Therefore, λl are the eigenvalues of a(v), and T (W ) = 0 on
all subspaces ker(a(v) − λl ) for λl 6= 1 Therefore, W (T ) is zero for any function
supported by Vµd \ (ker T )◦ .
Consider decompositions


ℓ2 (Vµd ) = ℓ2 (ker T )◦ ⊕ ℓ2 Vµd \ (ker T )◦ ;


ℓ2 (Vνd ) = ℓ2 (indef T )◦ ⊕ ℓ2 Vνd \ (indef T )◦ .

The operator W (T ) has the following block form with respect to this decomposition


f (T ) 0
W
,
W (T ) =
0
0
11

with a non-zero block


f (T ) : ℓ2 (ker T )◦ → ℓ2 (indef T )◦ .
W

The linear relation T determines a linear relation T ′ : dom T ⇒ im T . We
take the projection
dom T ⊕ im T → (dom T / ker T ) ⊕ (im T / indef T )
and the image Te of T ′ under this projection. Thus we get a linear relation
Te : dom T / ker T ⇒ im T / indef T . The spaces dom T / ker T , im T / indef T have
form V2κ . By construction, ker Te = 0, indef Te = 0. Therefore, Te is a graph of
a symplectic operator in V2κ .
f (T ) has the same commutation relations with the HeisenIt is easy to that W
e
berg group as W (T ). It remains to refer to Proposition 1.1.


3.5. One corollary from the previous Subsection. For a linear embedding B : Vµd → Vνd we define two operators
∗
σB
: ℓ2 (Vµd ) → ℓ2 (Vνd )

σB : ℓ2 (Vνd ) → ℓ2 (Vµd ),
in the following way
σB ϕ(x) = ϕ(xB);
(

ψ B −1 (y) ,
∗
σB ψ(y) =
0,

if y ∈ im B;
if y ∈
/ im B.

In fact, in the previous section we proved the following lemma:
Lemma 3.6 Any operator W (T ) can be decomposed as product of the form
W (T ) = W (g1 )λ∗B W (g2 )λC W (g3 ),
where g1 , g2 , g3 are elements of symplectic groups and B, C are appropriate
embeddings.
3.6. Gaussian operators are bounded. Consider a Gaussain operator
G(H, Q) given by (1.3). Consider projections π1 : H → Vµd ⊕0, π2 : H → 0⊕Vνd .
We represent H as a finite disjoint union of affine subspaces Zj such that π1 ,
π2 are injective on Zj . Consider operators
X

Exp Q(x, y) f (y).
Gj f (x) =
y: (y,x)∈Zj

Actually, forPeach x the sum consists of 0 or 1 element. Clearly, kGj k = 1, and
G(H, Q) = Gj .
3.7. Products of Gaussian operators.
12

Lemma 3.7 Let X be a finite-dimensional space, Y be a sum of a finite or
countable number of copies of F. Let Q be a quadratic form on X × Y . Consider
the sum
X

F (y) =
Exp Q(x, y) .
(3.1)
x∈X

Then there is a subspace Z ⊂ Y of codimension 6 dim X, a nonzero constant
c, and a quadratic form R on Z such that
(

c · Exp R(y) ,
if y ∈ Z;
F (y) =
0,
if y ∈
/ Z.
Proof. This follows from the following observation (see, e.g., [12], Sect.
9.2). Let X, U be finite-dimensional spaces over F of the same dimension.
Consider a quadratic form P (x) on X. Let
X
X

f (u) =
Exp P (x) +
u j yj .
x∈X

Then there is a subspace K ⊂ U and a quadratic form S on U such that
(

c · Exp S(u) ,
if u ∈ K;
f (u) =
.
(3.2)
0,
if u ∈
/K
We represent Q(x, y) as
Q(x, y) = Q(x, 0) +

X

xj lj (y) + Q(0, y),

j

where lj are linear forms on Y , and apply formula (3.2).



Lemma 3.8 A product of Gaussian operators is a Gaussian operator up to a
nonzero scalar factor.
Proof. Consider Gaussian operators
G(H, Q) : ℓ2 (Vµd ) → ℓ2 (Vνd ),

G(K, R) : ℓ2 (Vνd ) → ℓ2 (Vκd ).

We consider the set Z of all triples (u, v, w) ∈ Vµd ⊕Vνd ⊕Vκd , such that (u, v) ∈ H,
(v, w) ∈ K. The kernel of the product is given by
X


Exp Q(u, v) Exp R(v, w) .
N (u, w) =
v: (u,v,w)∈Z

We get a sum of the form (3.1). More precisely, consider the natural projection
Z → Vµd ⊕ Vκd . Denote by X its kernel (it is finite-dimensional), let Y be a
subspace complementary to the X. We apply Lemma 3.7 and obtain a Gaussian
expression for the kernel N .

13

Corollary 3.9 For any g ∈ Sp(V2µ ) operators W (g) are Gaussian.
Proof. Indeed, for generators of Sp(V2µ ) the operators W (g) are Gaussian,
see Subs. 2.5. Their products are Gaussian.

Lemma 3.10 For any perfect Lagrangian linear relation T , the operator W (T )
is Gaussian.
Proof. We refer to Lemma 3.6.
3.8. End of proof Theorem 1.6.



Lemma 3.11 Any Gaussian operator has the form W (T ).
Proof. Consider a Gaussian operator G(H; Q) : ℓ2 (Vµd ) → ℓ2 (Vνd ). Extend
the quadratic form Q to Vµd ⊕ Vνd in an arbitrary way. Represent Q as
Q(x, y) = Q(x, 0) + Q(0, y) +

X

skl xk yl .

Let C(x) be a quadratic form on a space Vκd . Denote by G[[C]] the operator
in ℓ2 (Vκd ) given by

G[[C]]ϕ(x) = Exp C(x) ϕ(x)

Recall (see Subs. 2.5, formula (2.4)), that such operators have a form W (g) for
certain g ∈ Sp(V2κ ). Consider a Gaussian operator
G := G[[−Q(x, 0)]] G[H; Q] G[[−Q(0, y)]].
Clearly, the statements ’the operator G(H, Q) has a form W (·)’ and ’the operator
G has a form W (·)’ are equivalent. The operator G has a form

X
X
X
Exp(xSy t ) ψ(y).
Exp
skl xk yl ψ(y) =:
Gψ(x) =
y: (y,x)∈H

y: (x,y)∈H

Let us describe the set T of all (v, w) ∈ V2µ ⊕ V2ν such that
a(w)G = Ga(v).
Consider (p, q) ∈ H. Then

(p, qS); (q, −pS) ∈ T.

Next, let (ξ, η) ∈ Vµc ⊕ Vνc be contained in H ◦ . Then

(0, ξ); (0, η) ∈ T.

(3.3)

(3.4)

It is easy to see that elements of forms (3.3) and (3.4) generate a perfect Lagrangian relation.

3.9. Products of perfect Lagrangian relations. Proof of Theorem
1.4.
14

Lemma 3.12 Fix a perfect Lagrangian relation T . Let (β, γ) satisfy
a(γ)W (T ) = W (T )a(β).
Then (β, γ) ∈ T .
Proof. Let p, q ∈ V2∞ . Then

a(p)a(q)a(−p)a(−q) = Exp {p, q} .

Let (β, γ) does not contained in T . Choose a vector (u, v) ∈ T that is not
orthogonal to (β, γ). This means that {β, u} 6= {γ, v}. For a field F of a prime
order this implies


Exp {β, u} 6= Exp {γ, v} .
(3.5)
For an arbitrary finite field we can multiply (u, v) by a constant factor, in this
way we can achieve (3.5). Next, we have
a(β)a(γ)a(−β)a(−γ)W (T ) = W (T )a(u)a(v)a(−u)a(−v),
or



Exp {β, u} · W (T ) = Exp {γ, v} · W (T )

Hence W (T ) = 0, and we came to a contradiction.



Lemma 3.13 Let T : V2µ ⇒ V2ν , S : V2ν ⇒ V2κ be perfect Lagrangian linear
relations. Then the linear relation ST satisfies the following properties:
a) The linear relation ST is isotropic and for any (v, w) ∈ ST we have
a(w) W (S) W (T ) = W (S) W (T ) a(v).

(3.6)

b) ker ST , indef ST are compact, and dom ST , im ST are codiscrete.
c) ST is contained in a certain perfect Lagrangian relation R such that
W (R) = W (S)W (T ).
Proof. a) Let (v, w), (v ′ , w′ ) ∈ ST . Choose u, u′ ∈ V2ν such that (v, u),
(v , u′ ) ∈ T and (u, w), (u′ , w′ ) ∈ S. Since T , S are isotropic, we have
′

{u, u′ } = {v, v ′ } = {w, w′ }.
Thus ST is isotropic.
Next,
W (S)W (T )a(v) = W (S)a(u)W (T ) = a(w)W (S)W (T ).
b) The subspace ker ST is the set of all v such that there is u satisfying
u ∈ ker T , (v, u) ∈ S. Thus we take the preimage Z of ker T under the projection
T → Vµ ⊕ 0, and ker ST is the projection of Z to 0 ⊕ Vµ . Fibers of the projection
T → Vµ ⊕ 0 are compact, therefore Z is compact, and ker ST is compact.
15

Let us verify the statement about im ST . The subspace im T ∩ dom S is
codiscrete in dom S. Its image H under the projection dom S → dom S/ ker S
also is codiscrete. The relation S determines a symplectic isomorphism Se :
e is codiscrete in im S/ indef S.
dom S/ ker S → im S/ indef S. The subspace SH
Therefore its lift to im S (it is im ST ) is codiscrete in im S and therefore in V2κ .
c) Operators W (T ) and W (S) are Gaussian. Therefore, W (S)W (T ) is Gaussian, and hence it has a form W (R).

Lemma 3.14 Let X, Y be codiscrete coisotropic subspaces in V2∞ . Consider
the symplectic space X/X ♦ , the image K of X ∩Y ⊂ X in X/X ♦ and the image
L of X ∩ Y ♦ ⊂ X in X/X ♦. Then L = K ♦ in X/X ♦. Moreover, L is compact,
and K is codiscrete.
Proof. We have
♦
e
e = (X ∩ Y ) + X ♦ ;
K = K/X
, where K
e ♦ , where L
e = (X ∩ Y ♦ ) + X ♦ .
L = L/X

The space X is equipped with a skew-symmetric bilinear form, whose kernel
e and L
e are orthogonal in X, therefore K and L are
is X ♦ . It is clear that K
orthogonal in X/X ♦. Next,
e ♦ = (X ♦ + Y ♦ ) ∩ X.
K

e ♦ . Let e
Let h ∈ K
h be its representative, e
h = a + b, where a ∈ X ♦ , b ∈ Y ♦ .
Then b also is a representative of h, and hence h ∈ L.

Lemma 3.15 Let T : V2µ ⇒ V2ν be a perfect Lagrangian linear relation. Let
R ⊂ T be a relation V2µ ⇒ V2ν . Assume that
(ker R)♦ = dom R,

(dom R)♦ = ker R,

(im R)♦ = indef R,

(indef R)♦ = im R.

or
Then T = R.
Proof. Let R ( T . Considering projection to V2µ ⊕ 0, we get
dom R ( dom T,

or

indef R ( indef T.

Therefore,
dom R ( (ker T )♦ ⊂ (ker R)♦ ,
or
indef R ( indef T = (im T )♦ ⊂ (im R)♦ .
This contradicts to the conditions of the lemma.
16

,

Proof of Theorem 1.4. Let T : V2µ ⇒ V2ν , S : V2ν ⇒ V2κ be perfect
Lagrangian linear relations. We wish to prove that ST is perfect Lagrangian.
Without loss of generality we can assume
ker T = 0,

indef S = 0.

(3.7)

Otherwise, we take the natural projection
dom T ⊕ V2ν → (dom T / ker T ) ⊕ V2ν
and the image Te of T under this projection. We get a linear relation Te :
dom T / ker T ⇒ V2ν . Clearly, the statements ’ST is perfect Lagrangian’ and
’S Te is perfect Lagrangian’ are equivalent. In the same way we can assume
indef S = 0.
Under the condition (3.7),
dom T = V2µ ,

im S = V2κ .

Next,
im ST = S im T = S(im T ∩ dom S),
indef ST = S indef T = S(indef T ∩ dom S).
Consider the map dom S ⊕V2κ → (dom S/ ker S)⊕V2κ . Let Sb be the image of S
under this map. It is a graph of an symplectic bijection σ : dom S/ ker S → V2κ .
We have
h
i

im ST = σ (im T ∩ dom S) + ker S / ker S ,
h
i

indef ST = σ (indef T ∩ dom S) + ker S / ker S .

By Lemma 3.14, the spaces in the square brackets are orthogonal complements
one to another. Therefore, the same holds for the left hand sides. By Lemma
3.15, ST is Lagrangian.

3.10. End of proof Theorem 1.5. After the establishment of Theorem
1.4 Lemma 3.13.a becomes the statement b) of Theorem 1.5.
To prove the statement c) of Theorem 1.5 we write adjoint operators to the
both sides of (1.2),
W (P )∗ a(−w) = a(−v)W (P )∗ ,
or
a(v)W (P )∗ = W (P )∗ a(w).
This is the defining relation for the operator W (P  ).

17

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Math.Dept., University of Vienna,
Oskar-Morgenstern-Platz 1, 1090 Wien;
& Institute for Theoretical and Experimental Physics (Moscow);
& Mech.Math.Dept., Moscow State University;
& Institute for information transmission problems (Moscow);
e-mail: neretin(at) mccme.ru
URL:www.mat.univie.ac.at/∼neretin

19

