By Tevelev E.

In the course of a number of centuries numerous reincarnations of projective duality have encouraged learn in algebraic and differential geometry, classical mechanics, invariant idea, combinatorics, and so forth. nevertheless, projective duality is just the systematic means of getting better the projective style from the setof its tangent hyperplanes. during this survey we've attempted to assemble togetherdifferent features of projective duality and issues of view on it. we are hoping, thatthe exposition is sort of casual and calls for just a typical wisdom ofalgebraic geometry and algebraic (or Lie) teams conception. a few chapters are,however, tougher and use the fashionable intersection conception and homologyalgebra. yet even in those instances now we have attempted to provide easy examples andavoid technical problems.

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R denote simple roots then each (i, j) ∈ S corresponds to a positive root αi + αi+1 + . . + αj . Another useful interpretation is to regard a pair (i, j) ∈ S as a segment [i, j] = {i, i + 1, . . , j} in Z. A family m = mij ∈ ZS+ can be regarded as a collection of segments, containing mij copies of each [i, j]. Thus, elements of ZS+ can be called multisegments. The weight |m| of a multisegment m is defined as a sequence γ = {d1 , . . , dr ) ∈ Zr+ given by di = mkl for i = 1, . . , r. i∈[k,l] In other words, |m| records how many segments of m contain any given number i ∈ [1, r].

These embeddings can be done equivariantly. Points of V ∗ correspond to hyperplanes in Pn not passing through the origin 0 ∈ V ⊂ Pn . Suppose that O ⊂ V ∗ is a non-conical orbit. Then the dual ∗ ∗ variety O ⊂ Pn intersects with Cn non-trivially. Therefore, O ⊂ Pn is the ∗ ∗ closure of a conical variety in Cn . Therefore, its dual variety O ⊂ Pn ∗ does not intersect (Cn )∗ . But this contradicts the Reflexivity Theorem. 8. 9 ([Py]) Suppose that a connected algebraic group G acts linearly on a vector space V with a finite number of orbits.

We associate to m the multisegment m given by m = m − (i1 , j1 ) − (i2 , j2 ) − . . − (ip , jp ) + (i2 , j1 ) + (i3 , j2 ) + . . + (ip+1 , jp ), where we use the convention that (i, j) = 0 unless 1 ≤ i ≤ j ≤ r. 12 ([MW]) If the multisegment m is associated to m then ζ(m) = ζ(m ) + (i1 , ip ). The involution ζ can also be described in terms of irreducible finitedimensional representations of affine Hecke algebras and in terms of canonical bases for quantum groups, see [KZ]. 3 Parabolic Subgroups With Abelian Unipotent Radical Let L be a simple algebraic group and P ⊂ L a parabolic subgroup with abelian unipotent radical.