OPEN SYMMETRIC ORBITS OF REDUCTIVE GROUPS IN SYMMETRIC<i>R</i>-SPACES

Makarevič, B O

doi:10.1070/sm1973v020n03abeh001882

Cited by 13 publications

(11 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The fact that they also admit globally a twisted para-complexification has been remarked in [Be98, 2.1, Ex.4]. The classification [Ma73] of irreducible Makarevič spaces essentially coincides with the classification of irreducible symmetric spaces with twist (Chapter 4); however, the close relation with Jordan triple systems and twisted complexifications was not remarked in [Ma73].…”

Section: Definition 227mentioning

confidence: 98%

Complexifications of symmetric spaces and Jordan theory

Bertram

2001

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevič in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems. Introduction Straight and twisted complexifications.The Euclidean vector space (R n , i x i y i ) can be "complexified" in two ways: first, we have the extension (C n , i z i w i ) of the scalar product to a complex bilinear form on C n , and second, we have the "Hermitification" (C n , i z i w i ) which yields a scalar product on C n . Of course, this works for any bilinear form b on R n ; let us call b C (z, w) its "straight" (complex bilinear) and b C (z, w) its "twisted" complexification.In this paper we investigate an analogue of this construction in a situation which geometrically is much less trivial due to the presence of curvature. A natural class of spaces to look at here is the class of symmetric spaces since these are characterized by one fundamental invariant, namely by the curvature tensor R itself which satisfies the algebraic identities of a Lie triple system (LTS; cf. Def. 1.1.1). It is almost trivial that any such space admits (locally) a straight complexification given by the C-trilinear extension R C of R; from a group theoretic point of view this can be interpreted as the complexification of the homogeneous symmetric space M = G/H locally by the complex symmetric space for the curvature R, the invariant almost complex structure J and vector fields X, Y, Z. We consider this relation as fundamental and take it as the defining relation of a twisted complex symmetric space. A twisted complexification is then a (local) imbedding of M as a real form of a twisted complex symmetric space. For instance, the unit disc SU(1, 1)/ SO(2) is a twisted complexification of its real form M = ] − 1, 1[ ∼ = R; the complex projective space U(n + 1)/(U(n) × U(1)) is a twisted complexification of the real projective space M = O(n + 1)/(O(n) × O(1)); the variety Gl(2n, R)/ Gl(n, C) of complex structures on R 2n is a twisted complexification of the symmetric space of group type M = Gl(n, R) (cf. Ex. 1.3.7).We do not claim that any symmetric space admits a twisted complexification and, if it does, that it is unique. In fact, there is a very subtle curvature obstruction conveniently described in terms of Jordan theory: it would be tempting to define the twisted complexification of a LTS R by R C (X, Y )Z, but this does not work. However, we will see that in the Jordan category this idea works perfec...

show abstract

Section: Definition 227mentioning

confidence: 98%

Complexifications of symmetric spaces and Jordan theory

Bertram

2001

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Notice that canonical overgroups exist for all 10 series of real classical groups. 3 Moreover overgroups exist for all 52 series of classical semisimple symmetric spaces G/H , see [8,15], see also [18], Addendum D.6. So the problem makes sense for all classical symmetric spaces.…”

Section: Plancherel Formula For the Restriction Of A Unitary Represenmentioning

confidence: 99%

The Fourier Transform on the Group $$\mathrm {GL}_2({\mathbb R })$$ GL 2 ( R ) and the Action of the

Neretin

2017

J Fourier Anal Appl

View full text Add to dashboard Cite

We define a kind of 'operational calculus' for the Fourier transform on the group GL 2 (R). Namely, GL 2 (R) can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in R 4 . Therefore the group GL 4 (R) acts in L 2 on GL 2 (R). We transfer the corresponding action of the Lie algebra gl 4 to the Plancherel decomposition of GL 2 (R), the algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of gl 4 ⊕ gl 4 in the Plancherel decomposition of GL 2 (C).

show abstract

“…We consider not only symmetric spaces G/H which can globally be written as a quotient G b /Q, but also spaces having "locally" such a realization; in the semisimple case these are the "open symmetric orbits in symmetric R-spaces" classified by B.O. Makarevič ([Ma73]). In our previous work [Be97] we have shown that algebraically such spaces, called symmetric spaces with twist, are equivalent to Jordan triple systems (JTS), and we have given an intrinsic 40 W. Bertram geometric characterization which is the starting point for the presentation given here.…”

mentioning

confidence: 99%

Conformal group and fundamental theorem for a class of symmetric spaces

Bertram

2000

Math Z

View full text Add to dashboard Cite

Using a geometric approach, we define and investigate the conformal group of a symmetric space with twist. In the non-degenerate case we characterize this group by a theorem generalizing the fundamental theorem of projective geometry. IntroductionThe real projective space RP n = O (n + 1)/( O (n) × O (1)) and the Sphere S n = SO(n + 1)/ SO(n) are examples of a class of symmetric spaces M = G/H which can also be written as a quotient M = G b /Q with respect to a "big Lie group" G b . In the first case, G b = P Gl(n + 1, R) is the general projective group, characterized by the fundamental theorem of projective geometry as the group preserving collinearity (if n > 1), and in the second case, G b = SO(n+1, 1) is the conformal group, characterized by a classical result of Liouville (1850) as the group preserving the Riemannian metric of the sphere up to a scalar function (if n > 2). Another example is the unitary group U (n) which is homogeneous under the "big group" SU(n, n). Here I.E. Segal offered the conjecture that the "big group" can be characterized as the causal group of U (n) ([Se76, p.35]).In this work we present a unified theory of the "big group". We consider not only symmetric spaces G/H which can globally be written as a quotient G b /Q, but also spaces having "locally" such a realization; in the semisimple case these are the "open symmetric orbits in symmetric R-spaces" classified by B.O. Makarevič ([Ma73]). In our previous work [Be97] we have shown that algebraically such spaces, called symmetric spaces with twist, are equivalent to Jordan triple systems (JTS), and we have given an intrinsic 40 W. Bertram geometric characterization which is the starting point for the presentation given here.Every symmetric space with twist admits a twisted complexicification; for example, if M = RP n , then the twisted complexification is the complex projective space CP n . In the present work we characterize the twisted complex spaces as circled spaces; these are complex manifolds with a real affine connection such that multiplication by i in each tangent space extends to a holomorphic affine (local) diffeomorphism (Chapter 1). By a sort of analytic continuation, we deduce that multiplication by real scalars in each tangent space also has a canonical extension to a local diffeomorphism (no longer affine). The corresponding local one-parameter group attached to each point p in the manifold induces a vector field E = E p which we call an Euler operator on a symmetric space. In the example of M = RP n , using the usual affinization V = R n of M , the Euler operator E = E 0 is nothing but the usual Euler operator E(x) = x on R n . In fact, to any Euler operator E p on a symmetric space we canonically associate a chart, called Jordan coordinates, such that E p is represented by the usual Euler operator (Chapter 2). Now, if M = G/H, then the Lie algebra g of G together with the Euler operators generate a "big Lie algebra" g b of vector fields on M , called the (inner) conformal Lie algebra. It is worth noticing that, becaus...

show abstract

OPEN SYMMETRIC ORBITS OF REDUCTIVE GROUPS IN SYMMETRICR-SPACES

Cited by 13 publications

References 3 publications

Complexifications of symmetric spaces and Jordan theory

Complexifications of symmetric spaces and Jordan theory

The Fourier Transform on the Group $$\mathrm {GL}_2({\mathbb R })$$ GL 2 ( R ) and the Action of the

Conformal group and fundamental theorem for a class of symmetric spaces

Contact Info

Product

Resources

About