Journal of Mathematical Sciences, Vol. 138, No. 3, 2006

ON COMPRESSION OF BRUHAT–TITS BUILDINGS

Yu. A. Neretin∗ UDC 512.54, 514.7

Consider an affine Bruhat–Tits building Latn of type An−1 and the complex distance in Latn, i.e., the completesystem of invariants of a pair of vertices of the building. An element of the Nazarov semigroup is a lattice in theduplicated p-adic space Qn

p ⊕ Qnp . We investigate the behavior of the complex distance with respect to the natural

action of the Nazarov semigroup on the building. Bibliography: 18 titles.

It is well known that affine Bruhat–Tits buildings (see, for instance, [1, 3, 7]) are right p-adic analogs ofRiemannian noncompact symmetric spaces. This analogy has no a priori explanations, but it exists at the levelsof geometry, harmonic analysis, and special functions. Some known examples of this parallel are

• Gindikin and Karpelevich’s and Macdonald’s Plancherel formulas, see, e.g., [7];• the existence of a general theory including the Hall–Littlewood polynomials and zonal spherical functions,

see [8];• similarity in geometries at infinity (Semple–Satake type boundaries, Martin boundary, etc.), see [4, 13];• the parallel between the Horn–Klyachko triangle inequality (see [2]) for the eigenvalues of Hermitian

matrices1 and the triangle inequality for the complex distance in buildings;2

• the existence of a precise parallel between the harmonic analysis of the Berezin kernels and of the matrixbeta functions at the level of buildings,3 see [16].

In this note, we continue this parallel and obtain a (quite simple) analog of the theorem on compression ofangles (this theorem appeared in [11] as a lemma on the boundedness of some integral operators in Fock spaces;see its proof in [12, 6.3] and some geometrical discussion in [14]; see also [15] for its extension to exceptionalgroups) for buildings of type An.

1. Formulation of the result

1.1. Notation. We denote by Qp the p-adic field, and by | · | the norm in Qp. Let Op be the ring of p-adicintegers. Recall that Op consists of elements with norm � 1. The coordinate n-dimensional linear space over Qp

will be denoted by Qnp .

By GLn(Qp) we denote the group of invertible n×n matrices over Qp, and by GLn(Op) the group of matricesg ∈ GLn(Qp) such that all matrix elements of g and g−1 are integers.

1.2. The space of lattices. A lattice in Qnp is a compact open Op-submodule. Recall that any lattice R can

be represented in the formR = Of1 ⊕ · · · ⊕ Ofn, (1)

where f1, . . . , fn is some basis in Qnp ; see [18, § II.2]. We denote by Latn the space of lattices in Qn

p (this spaceis countable, and the natural topology on Latn is discrete).

The group GLn(Qp) acts transitively on Latn. The stabilizer of the lattice Onp is GLn(O). Thus

Latn = GLn(Qp)/ GLn(Op).

1.3. The complex distance.

∗Institute for Theoretical and Experimental Physics, Moscow, Russia, and University of Vienna, Austria,

e-mail: neretin@mccme.ru.1The interpretation of this inequality in terms of the geometry of symmetric spaces is discussed in [15].2Let R, S, T ∈ Latn (see the notaion below). The problem is to describe possible collections of 3n integers {kj(R,S)}, {kj (R,T )},

and {kj(S, T )}. It can easily be reduced to the following (unexpectedly nontrivial) question solved in the 1960s by Ph. Hall and

T. Klein. Let A and B be finite Abelian p-groups. For which Abelian p-groups C there exists a subgroup A′ isomorphic to A such

that C/A′ � B; for a solution, see [8, II.3].3The matrix gamma function corresponds to the Tamagawa zeta function, see [8, V.4].

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 163–170. Original article submitted September28, 2004.

5722 1072-3374/06/1383-5722 c©2006 Springer Science+Business Media, Inc.

Theorem (see, for instance, [18, Theorem 2.2.2]). Let R and S be two lattices in Qnp .

(a) There exists a basis f1, . . . , fn ∈ Qnp such that

R =n∑

j=1

Opfj , S =n∑

j=1

p−kj(R,S)Opfj . (2)

The integer numbers

k1(R, S) � k2(R, S) � . . . � kn(R, S)

are uniquely determined by the lattices R and S.

(b) If S, R and S′, R′ are elements of Latn satisfying

kj(R, S) = kj(R′, S′),

then there exists g ∈ GLn(Qp) such that

R = g · R′, S = g · S′.

We say that kj(R, S) is the complex distance between S and R (this structure is an analog of the complexdistance or stationary angles in symmetric spaces; see, for instance, [15]).

1.4. The product of relations. Let V and W be sets. A relation H : V ⇒ W is a subset in V ×W .Let H : V ⇒ W and G : W ⇒ Y be relations. Their product G · H : V ⇒ Y is the relation consisting of all

pairs v × y ∈ V × Y for which there exists w ∈ W satisfying v × w ∈ H, w × y ∈ G.

1.5. The Nazarov semigroup. An element of the Nazarov semigroup Γn is a lattice in Qnp ×Qn

p . The productin the Nazarov semigroup is the product of relations.

Remark. One may regard elements of the group GLn(Qp) as points of the Nazarov semigroup at infinity.Indeed, for g ∈ GLn(Qp) consider a sequence of lattices Zj ⊂ Qn

p ⊕ Qnp such that

∩∞j=1 ∪∞

k=j Zj

coincides with the graph of g. For every lattice R ⊂ Qnp , we have Zj · R = g ·R starting from some j.

Remark. Nazarov semigroups are p-adic analogs of semigroups of linear relations (see [12]). The main nontrivialproperty of Nazarov semigroups is given by the following assertion. The Weil representation of Sp(2n, Qp) (see[17]) admits a canonical extension to a representation of a Nazarov semigroup related to buildinds of type Cn; see[10, 9]. It seems that this assertion must have many corollaries (due to the opportunities of the Howe duality);these corollaries were never discussed in the literature.

1.6. The action of the Nazarov semigroup on Latn. Let H ∈ Γn, R ∈ Latn. We define HR ∈ Latn as theset of all w ∈ Qn

p such that there exists v ∈ R satisfying v × w ∈ H.

1.7. The result of the paper. The main result of this paper is the following theorem.

Theorem. Let H be an element of the Nazarov semigroup Γn. Let R, S ∈ Latn. Then for each j the numberkj(H · R, H · S) lies between 0 and kj(R, S), i.e.,

kj(R, S) � 0 implies 0 � kj(H ·R, H · S) � kj(R, S),

kj(R, S) � 0 implies 0 � kj(H ·R, H · S) � kj(R, S).

Remark. In the case of Riemannian symmetric spaces, there exists a converse theorem on the characterizationof all angle-compressive maps [14]. I do not know such assertions in the p-adic case.

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2. Proof

2.1. Norms. A norm on Qnp is a function v �→ ‖v‖ from Qn

p to the set

{. . . , p−1, 1, p, p2, . . .}

satisfying the conditions(1) ‖v + w‖ � max(‖v‖, ‖w‖);(2) ‖λv‖ = |λ|‖v‖ for λ ∈ Qp, v ∈ Qn

p .The set of all v such that ‖v‖ � 1 is a lattice.Conversely, let L be a lattice. We define the corresponding norm ‖ · ‖L by the rule

‖v‖L :=(

mink: pkv∈L

pk)−1

.

2.2. The minimax characterization of the complex distance. The following assertion is an exact analogof the standard minimax characterization of eigenvalues (see, for instance, [6]) and stationary angles (see, forinstance, [15]). The proofs are also similar.

Lemma 1. For L, M ∈ Latn,

(a) pkj(R,S) = maxW : dim W=j, W⊂Qn

p

{min

v∈W, v �=0

‖v‖S

‖v‖R

};

(b) pkj(R,S) = minW : W⊂Qn

p , codim W=j−1

{max

v∈W, v �=0

‖v‖S

‖v‖R

}.

Proof. Let R and S have the form (2). Consider the subspace Y = Qpfj ⊕· · ·⊕Qpfn. We have ‖v‖R/‖v‖S � pkj

for every v ∈ Y .If dim Wj = j, then Y ∩W = 0, whence minv∈W ‖v‖R/‖v‖S � pkj .For W = Qpf1 ⊕ · · · ⊕ Qpfj, we have minv∈W ‖v‖R/‖v‖S = pkj . �

2.3. Another characterization of the complex distance. The following lemma is obvious.

Lemma 2. Let L, M ∈ Latn. Then we have the following isomorphisms of finite Abelian p-groups:

M/(L ∩ M) =⊕

j: kj(L,M)>0

Z/pkj(L,M)Z,

L/(L ∩ M) =⊕

j:kj(L,M)<0

Z/p−kj(L,M)Z.

2.4. Duality and the complex distance. Denote by (Qnp )� the linear space dual to Qn

p . For a latticeL ∈ Latn, we consider the dual lattice L� ⊂ (Qn

p )� consisting of linear functionals f ∈ (Qnp )� such that

v ∈ L =⇒ f(v) ∈ Op.

Lemma 3. kj(L, M) = kj(M�, L�).

2.5. The complex distance and intersections of lattices.

Lemma 4. Let L, M , N ∈ Latn. Then for all j, the number kj(L ∩ M, L ∩ N) lies between 0 and kj(M, N).

This assertion is a corollary of Lemma 1 and the following Lemma 5.

Lemma 5. For v ∈ Qnp , the number ‖v‖L∩M/‖v‖L∩N lies between 1 and ‖v‖M/‖v‖N .

Proof. Denotelogp ‖v‖L = l, logp ‖v‖M = m, logp ‖v‖N = z.

Thenlogp ‖v‖L∩M = max(l, m), logp ‖v‖L∩N = max(l, z).

It remains to observe that max(l, m) − max(l, z) lies between 0 and l − z. �

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2.6. The complex distance and sums of lattices.

Lemma 6. Let L, M , N ∈ Latn. Then kj(L + M, L + N) lies between 0 and kj(M, N).

Proof. For P , Q ∈ Latn, we have(P + Q)� = P � ∩ Q�.

By duality considerations, our lemma is equivalent to Lemma 4. �2.7. Notation. Let S ⊂ Qn

p ⊕ Qnp be an element of the Nazarov semigroup. We define

– the kernel Ker S = S ∩ (Qnp ⊕ 0),

– the domain DomS as the image of S under the projection Qnp ⊕ Qn

p → Qnp ⊕ 0,

– the image ImS as the image of S under the projection Qnp ⊕ Qn

p → 0 ⊕ Qnp ,

– the indefiniteness Indef S as the intersection S ∩ 0 ⊕ Qnp .

Obviously, all the sets DomS, Ker S, Indef S, and Im S are lattices in Qnp .

2.8. Proof of the theorem. Let S ∈ Γn. We have

S ⊂ DomS ⊕ Im S, S ⊃ Ker S ⊕ Indef S.

Consider the (finite) quotient group

Y :=(DomS/ Ker S

)⊕

(ImS/ Indef S

)

and the image σ of S in Y . It can easily be checked that σ is the graph of an isomorphism

DomS/ Ker S → ImS/ Indef S.

In particular, these two groups are isomorphic.Hence, by Lemma 2,

kj(DomS, Ker S) = kj(Im S, Indef S).

This implies the existence of a (noncanonical) element gS ∈ GLn(Qp) such that

gS · DomS = Im S, gS · Ker S = Indef S.

For any L ∈ Latn, we haveS · L = gS · (L ∩ DomS) + Indef S.

It remains to apply Lemmas 4 and 6. �

Translated by Yu. A. Neretin.

REFERENCES

1. K. Brown, Buildings, Springer-Verlag, New York (1989).2. W. Fulton, “Eigenvalues of sums of Hermitian matrices (after A. Klyachko),” Asterisque, No. 252, Exp.

No. 845, 5, 255–269 (1998).3. P. Garrett, Buildings and Classical Groups, Chapman & Hall, London (1997).4. Y. Guivarch, L. Ji, and J. Taylor, Compactifications of Symmetric Spaces, Birkhauser Boston, Boston (1998).5. Kh. Koufany, “Contractions of angles in symmetric cones,” Publ. Res. Inst. Math. Sci., 38, No. 2, 227–243

(2002).6. V. B. Lidskii, “Inequalities for eigenvalues and singular values,” Addendum to the book F. R. Gantmaher,

Theory of Matrices, 2nd (1966), 3rd (1976), 4th (1988) Russian editions.7. I. G. Macdonald, Spherical Functions on a Group of p-Adic Type, Publ. Ramanujan Inst., Madras (1971).8. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford (1995).9. M. L. Nazarov, “The oscillator semigroup over nonarchimedian field,” J. Funct. Anal., 128, 384–438 (1995).

10. V. Nazarov, Yu. Neretin, and G. Olshanski, “Semi-groupes engendres par la representation de Weil du groupesymplectique de dimension infinie,” C. R. Acad. Sci. Paris Ser. I Math., 309, No. 7, 443–446 (1989).

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11. Yu. Neretin, “On a semigroup of operators in the boson Fock space,” Funct. Anal. Appl., 24, No. 2, 135–144(1990).

12. Yu. Neretin, Categories of Symmetries and Infinite-Dimensional Groups, The Clarendon Press. Oxford Uni-versity Press, New York (1996).

13. Yu. A. Neretin, “Hinges and the Study–Semple–Satake–Furstenberg–De Concini–Procesi–Oshima boundary,”in: Kirillov’s Seminar on Representation Theory, Amer. Math. Soc. Transl. Ser. 2, 181, Amer. Math. Soc.,Providence, Rhode Island (1998), pp. 165–230.

14. Yu. A. Neretin, “Conformal geometry of symmetric spaces, and generalized linear-fractional Kreın–Shmulianmappings,” Sb. Math., 190, No. 1–2, 255–283 (1999).

15. Yu. A. Neretin, “Jordan angles and triangle inequality in Grassmannian manifold,” Geometria Dedicata, 86,403–432 (2001).

16. Yu. A. Neretin, “The Beta function of the Bruhat–Tits building and the deformation of the space L2 on theset of p-adic lattices,” Sb. Math., 194, No. 11–12, 1775–1805 (2003).

17. A. Weil, “Sur certains groupes d’operateurs unitaires,” Acta Math., 111, 143–211 (1964).18. A. Weil, Basic Number Theory, Springer-Verlag, New York (1967).

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