Representations and binomial coefficients

To a root system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> and a choice of coefficients in a field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> we associate a category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒳</mml:mi> </mml:math> of graded spaces with operators . For an arbitrary choice of coefficients we show that we obtain a semisimple category in which the simple objects are parametrized by their highest weight. Then we assume that the coefficients are given by quantum binomials associated to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> is an invertible element in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> . In the case that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> is simply laced and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> has positive (quantum) characteristic, we construct a Frobenius pullback functor and prove a version of Steinberg’s tensor product theorem for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒳</mml:mi> </mml:math> . Then we prove that one can view the objects in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒳</mml:mi> </mml:math> as the semisimple representations of Lusztig’s quantum group associated to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> we obtain semisimple representations of the hyperalgebra associated to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ). Hence we obtain new proofs of the Frobenius and Steinberg theorems both in the representation theory of reductive algebraic groups and of quantum groups.