Generalized binomial expansion on Lorentz cones

Transcription

1 J. Math. Anal. Appl Generalized binoial expansion on Lorentz cones Honging Ding Departent of Matheatics and Matheatical Coputer Science, University of St. Louis, St. Louis, MO 6303, USA Received 3 August 00 Subitted by D. Wateran Abstract We study soe properties of generalized binoial coefficients for syetric cones and we obtain a generalized binoial expansion forula for Lorentz cones. 003 Elsevier Science USA. All rights reserved. Keywords: Jordan algebra; Syetric cone; Lorentz cone; Spherical polynoial; Spherical function; Binoial expansion. Introduction Let Ω be an irreducible syetric cone and let J be the associated Jordan algebra. Denote the rank of Ω by r and the diension of J by n. Fix a Jordan frae {e,...,e r } in J and define the following subspaces: V j {x J : e j x x} and V ij {x J : e i x /x and e j x /x}, where is the Jordan product in J.ThenV j Re j for j,...,r are -diensional subalgebras of J, while the subspaces V ij of J for i, j,...,r with i j all have a coon diension d. It follows that n r + d/rr and n r + d r.. For j,...,r,lete j e + +e j,andsetj j {x J : E j x x}.thenj j is a subalgebra of J of rank j. Denote by P j the orthogonal projection of J onto J j and define j x δ j P j x. This research was supported in part by USA National Science Foundation Grant DMS E-ail address: dingh@slu.edu X/03/$ see front atter 003 Elsevier Science USA. All rights reserved. doi:0.06/s00-47x

2 H. Ding / J. Math. Anal. Appl for x J,whereδ j denotes the Koecher nor, or the deterinant function for J j.then j is a polynoial on J that is hoogeneous of degree j. We call j x the jth principal inor of x. Let λ,...,λ r be coplex nubers, and define the function λ on J by r λ x x λ r j x λ j λ j+,.3 j where x r x is the Koecher nor function on J. The function λ is the generalized power function on J. In particular, when λ j j are integers for all j,...,r and r 0,,..., r is called a partition, and we write 0. The length of is defined by + + r,and becoes a polynoial function on J which is hoogeneous of degree. LetGΩ denote the autoorphis group of Ω and let G be the connected coponent of the identity in GΩ. ThenG acts transitively on Ω G/K, wherek is the stability group of identity eleent e in Ω. In fact, K is actually a axial copact subgroup of G. For each partition, the spherical polynoial of weight on Ω ay be defined by Φ x k xdk,.4 K where dk is the Haar easure on K. The algebra of all K-invariant polynoials on J, denoted by PJ K, decoposes as PJ K CΦ. Let ρ ρ,...,ρ r be an r-tuple given by ρ j d/4j r for j,...,r,and define the spherical function φ λ on Ω for λ C r by φ λ x λ+ρ k xdk..5 K It is clear fro.4 and.5 that Φ and φ λ are K-invariant and for any partition, Φ φ ρ. Moreover, φ λ φ λ if and only if there exists a perutation w such that λ wλ. For a fixed Jordan frae {e,...,e r },anyx in J can be written as r x k a, k K, a a j e j,.6 j where a,...,a r are called the eigenvalues of x, and we ay assue a a r for the uniqueness. If x Ω, thenalla,...,a r are positive. Since the functions Φ and φ λ are K-invariant, they depend only on the eigenvalues a,...,a r of x Ω, andweay write Φ x Φ a,...,a r.letj C be the coplexification of J. Then every eleent z J C has a spectral decoposition z u a e + +a r e r,.7

3 68 H. Ding / J. Math. Anal. Appl where u U, a copact subgroup of GLJ C, a a r 0. z a is called the spectral nor of z J C. The open unit ball D of J C is defined by D { z J C : z < }..8 The generalized binoial expansion on the syetric cone Ω is [4, p. 343] Φ e + x Φ x,.9 0 where is a partition. Here, the binoial coefficients on syetric cones Ω are generalizations of the usual binoial coefficients n! n! n! which arises in the expansion + x x n.0 n n0 on the real line. It is known [5,] that 0 unless ;i.e.,if,..., r and,..., r,then j j for j,...,r. Therefore, the su in.9 has only finitely any ters Φ x for which. Because of iportance in analysis and other atheatical areas, generalized binoial coefficients have brought attentions to atheaticians; e.g., they are discussed in [5 9,]. By the spherical Taylor forula [4, XII.], d Φ Φ e + x n/r x x0 d Φ Φ n/r x xe x,. where d is the diension of the space P of polynoials on J generated by.in [3, Lea 4.4], we generalized the binoial coefficient for a partition to λ for λ C r. In [3, Proposition 4.5], we gave an integral expression for this binoial coefficient cf..6 below. However, still very little is known about the explicit for of these coefficients. By classification, there are four failies of classical irreducible syetric cones Π r R, ΠC, Π r H, the cones of all r r positive definite atrices over R, C, andh, the Lorentz cones Λ n, and an exceptional cone Π 3 O [4]. In [], we found an explicit binoial expansion forula for syetric cones Π r C and coputed the spherical transfor of φ λ I r + x for Π r C,whereI r is the r r identity atrix. In this paper, we study soe properties of the binoial coefficients for syetric cones Ω and we obtain an explicit binoial expansion for Lorentz cones Λ n. These results would be interesting for analysis on syetric cones, in particular, for analysis on Lorentz cones.

4 H. Ding / J. Math. Anal. Appl Properties of the binoial coefficients for syetric cones Let λ λ,...,λ r C r, λ x be the generalized power function defined by.3, d x x n/r dx,wheredx is the Lebesgue easure on Ω,andlettrx be the trace of x. The gaa function Γ Ω for the cone Ω is defined by Γ Ω λ e trx λ x d x. Ω whenever the integral converges absolutely. By [4, Theore VII..], the integral in. converges absolutely if and only if Re λ j >j d/ forj,,...,r. In this range, Γ Ω λ can be calculated by Γ Ω λ π n r/ r j Γ λ j j d,. and. defines the eroorphic continuation Γ Ω to all of C r. Note that /Γ Ω λ is an entire function on C r. Recall that for α C and nonnegative integer j, the classical Pochhaer sybol α j is defined by Γα+ j α j αα +...α+ j..3 Γα For λ C r and, any partition the Pochhaer sybol [λ] for Ω is defined by [λ] Γ Ωλ +..4 Γ Ω λ It follows fro. that r [λ] λ j j d. j j For,..., r Cr, a partition,..., r, and a positive integer k, define + k + k,..., r + k and + k + k,..., r + k. Thenwehave Theore.. + k + k kr [ k + ρ] k..5 [n/r + ] k Proof. By [3, Proposition 4.5],!Γ Ω + ρ Ω e trx d! [n/r] Φ xφ ρx d x.6

5 70 H. Ding / J. Math. Anal. Appl for Re j ρ j <d/4r, j,...,r. Siilarly, if Re j + k ρ j <d/4 r, j,...,r,then + k +k + k Γ Ω k + ρ e trx d +k Φ +k xφ [n/r] +k ρx d x..7 +k Ω It is easy to see that + k +kr. Since +k x x x k,and x is invariant under the subgroup K, Φ +k x Φ x x k..8 By the sae reason, φ +k ρx φ ρx x k φ ρx x k..9 By a forula for d [4, p. 35], d +k d. It follows fro.6.9 that + k kr Γ Ω + ρ[n/r] + k Γ Ω..0 k + ρ[n/r] +k By.0,.4, and a calculation,.5 holds for Re j + k ρ j <d/4r, j,...,r.since is a polynoial in C r [3, Lea 4.4],.5 holds for all C r. We now fix d,andlet,..., r Cr and,..., r be a partition. Theore.. The binoial coefficients satisfy,..., r, 0,..., r, 0 and,..., r, 0 0,..., r, r+ if r+ 0. Proof. Let Ω be an irreducible syetric cone of rank r + andletx,...,x r,x r+ be eigenvalues of x Ω defined by.6. By [0, Theore 5.3], Φ,..., r,0 x,...,x r,x r+ Γdr+ / x r x Γ d/ r+ x i x j d... Φ µ,...,µ r i<j x r+ x r r+ µ i x j d/ i µ p dµ...dµ r. i<pµ i j

6 H. Ding / J. Math. Anal. Appl for a partition,..., r. We rewrite the r-tuple ρ as ρr ;i.e.,ρ r ρ r,...,ρr r, where ρj r d/4j r for all j,...,r, and we introduce an r + -tuple ρr+ ; i.e., ρ r+ ρ r+,...,ρr r+,ρ r+ r+,whereρr+ j d/4j r.sinceφ φ ρ, we ay rewrite. as φ ρ r+,..., r ρr+ r, ρ r+ r+ x,...,x r,x r+ x r Γdr+ / Γ d/ r+ x i x j d... φ ρ r µ,...,µ r i<j x r+ x r r+ µ i x j d/ i µ p dµ...dµ r.. i<pµ i j By the generalized Carlson s theore [3, Corollary 3.],. holds for all C r. The proof of this fact is siple and is alost identical to the proof of [, Lea ]. It follows that φ ρ r+,..., r ρr+ r, ρr+ r+ + x,..., + x r, + x r+ x Γdr+ / + xi Γ d/ r+ + x j d i<j +x r +x r+... +x r r+ µi + x j d/ i<pµ i µ p dµ...dµ r i j x r x +x φ ρr µ,...,µ r Γdr+ / Γ d/ r+ x i x j d... φ ρ r + t,..., + t r i<j x r+ x r r+ t i x j d/ i t p dt...dt r..3 i<pt i j Theore. follows now fro Binoial expansion for Lorentz cones The only cone of rank is R + and Φ x x.if C,then + x x... + x 3.! 0 0 holds in the interval <x<. In particular, if is a positive integer, then the series 3. has only finitely any ters and it becoes.0. It is known [4] that the rank of a Lorentz cone Λ n is, the diension is n, d n, ρ d/4, and ρ d/4. The associated Jordan algebra with Λ n is J R n,andthe

7 7 H. Ding / J. Math. Anal. Appl coplexification J C of J is C n. As a special case of.7, every eleent z J C has a spectral decoposition z u a e + a e, 3. where {e,e } is a fixed Jordan frae, u U SOn SO, a a 0. By [4, Theore XII..], Φ z a a 3.3 for z given by 3.. By Theore., if is a positive integer, then Φ,0 + x, + x More generally, 0 φ +d/4, d/4 + x, + x Φ,0x,x Φ,0x,x, 3.5 where C, x and x are eigenvalues of x D. The expansions 3.4 and 3.5 are the binoial expansions for the special index,,where 0. For the general case 0, we have Theore 3.. For, C and a nonnegative integer, the binoial coefficients,, 0 k0 Proof. By.5 and.3, φ +d/4, d/4 + x, + x k k k ik d + i d + i. 3.6 φ +d/4, d/4 + x, + x e + x, 3.7 where x x 0 are two eigenvalues of x Λ n. By [4, Proposition XI.5.], e + x + Φ,0 x + Φ, x. By 3.3, Φ,0 x + Φ, x < in a neighborhood D of 0 in J C. By the binoial forula on R, e + x + Φ,0 x + Φ, x j Φ j,0 x j + ters involving Φ, x 3.8 for x D. By 3.5, j0 φ +d/4, d/4 + x, + x k 0 k Φ k,0 x 3.9

8 H. Ding / J. Math. Anal. Appl for x D. We now ultiply 3.8 with 3.9 and observe that the product of any ter involving Φ, x in 3.8 with any ter in 3.9 is a linear cobination of such spherical polynoials Φ, x for which 0. Hence, to study the binoial coefficient,,0, we need only to ultiply j0 j Φ j,0 x j with 3.9. By the recurrence forula [], Φ,0 xφ k,0 x d + k d + k Φ k+,0x + for a positive integer k. By atheatical induction, k d + k Φ k,x Φ,0 x k+j j d + i Φk,0 x Φ k+j,0 x d + i ik + ters involving Φ, x for which > To copute,,0, we ultiply j0 j j Φ,0 x j with 3.9 and apply 3.0. Adding all ters with k + j up and noticing j k, we prove 3.6. We are now ready to copute all binoial coefficients,, and derive the binoial forula for the Lorentz cones Λ n. By.5,,, [ + ρ],, 0 [n/ +, 0], d/, 0 + d/ +!. 3. By 3.6 and 3.,,, d/ + d/ +! k0 k k k d + i d + i. 3. ik

9 74 H. Ding / J. Math. Anal. Appl Theore 3. Binoial forula for Lorentz cones. φ +d/4, d/4 e + x, Φ, x, where the su is over all partitions, and the coefficients,, are given by 3.. Moreover, the series in 3.3 converges in D defined by 3., and defines an extension of φ +d/4, d/4 e + x to D.If, is a partition, then the series is finite. Proof. If, is a partition, then 3.3 is just.9, the coefficients are given by 3., and it is known that the series 3.3 is finite. It follows fro 3. that the coefficients,, have polynoial growth in. In the spectral decoposition 3. of z D, a a <. By 3.3, the series 3.3 converges in D. By., the both sides of 3.3 have the sae derivatives at 0, therefore, being analytic, they are equal in Λ n D. Moreover, the series in 3.3 defines an extension of φ +d/4, d/4 e + x to D. Acknowledgent The author would like to thank the referees for their expert and detailed coents. References [] H. Ding, Generalized binoial expansion on coplex atrix space, J. Math. Anal. Appl [] H. Ding, Raanujan s aster theore for Heritian syetric spaces, Raanujan J [3] H. Ding, K. Gross, D. Richards, Raanujan s aster theore for syetric cones, Pacific J. Math [4] J. Faraut, A. Koranyi, Analysis on Syetric Cones, Clarendon, Oxford, 994. [5] M. Lassalle, Une forule du binôe généralisée pour les polynôes de Jack, C. R. Acad. Sci. Paris Sér. I [6] M. Lassalle, Coefficients binoiaux généralisés et polynôes de Macdonald, J. Funct. Anal [7] A. Okounkov, G. Olshanski, Shifted Jack polynoials, binoial forula, and applications, Math. Res. Lett [8] A. Okounkov, G. Olshanski, Shifted Schur functions II. The binoial forula for characters of classical groups and its applications, Aer. Math. Soc. Transl. Ser [9] S. Sahi, The spectru of certain invariant differential operators associated to Heritian syetric spaces, in: J.L. Brylinsk Ed., Lie Theory and Geoetry, in: Progress in Math., Vol. 3, Birkhäuser, Boston, 994, pp

10 H. Ding / J. Math. Anal. Appl [0] P. Sawyer, Spherical functions on syetric cones, Trans. Aer. Math. Soc [] L. Vretare, Eleentary spherical functions on syetric spaces, Math. Scand [] Z. Yan, A class of generalized hypergeoetric functions in several variables, Canad. J. Math