Spherical Nilpotent Orbits and Unipotent Representations
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چکیده
then the coadjoint orbits of SL (2,R) fall into three basic classes; according to whether the Casimir function B (Z,Z) = h + xy is positive, negative or zero.(Here Z ≡ x ∗X + h ∗H + y ∗ Y .) The nature of these orbits becomes a little clear if we adopt a basis for which B is diagonal, setting Z0 = X − Y Z2 = X + Y Z3 = H we find B (Z,Z) = −z 0 + z 2 + z 3 That is, the invariant bilinear form on g looks like the Lorentz metric on R. And, in fact, the orbit structure of g looks like that of (2+1)-dimensional Minkowki spacetime; thinking of z0 as the “temperal coordinate” and z1 and z2 as the “spatial coordinates”. The three orbit classes are
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