By George Lusztig

ISBN-10: 0691081549

ISBN-13: 9780691081540

In this e-book Professor Lusztig solves an enticing challenge through fullyyt new equipment: in particular, using cohomology of constructions and comparable complexes.

The e-book provides an specific building of 1 exceptional member, D(V), of the discrete sequence of GL_{n} (F_{q}), the place V is the n-dimensional F-vector area on which GL_{n}(F_{q}) acts. it is a p-adic illustration; extra accurately D(V) is a unfastened module of rank (q--1) (q^{2}--1)...(q^{n-1}--1) over the hoop of Witt vectors W_{F} of F.

In bankruptcy 1 the writer reviews the homology of partly ordered units, and proves a few vanishing theorems for the homology of a few partly ordered units linked to geometric buildings. bankruptcy 2 is a research of the illustration ? of the affine team over a finite box. In bankruptcy three D(V) is outlined, and its limit to parabolic subgroups is decided. In bankruptcy four the writer computes the nature of D(V), and exhibits the right way to receive different participants of the discrete sequence by way of using Galois automorphisms to D(V). functions are in bankruptcy five. As one of many major purposes of his learn the writer provides an actual research of a Brauer lifting of the traditional illustration of GL_{n}(F_{q}).

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**Extra resources for Discrete Series of GLn Over a Finite Field. (AM-81)**

**Example text**

S by the formula Here S: is any leaf of *). Then a(E') = = E' C E" a(E") hence a(E') S a(E"). Assume next that E', E" E S1(E) \ S1v(E, ). Let S: be a leaf meeting E'. We have a(E') = < E', f > C < E", S: > = a(E"), hence a(E') S a(E"). Finally, assume that E' E S1(E) \ S1v(E, ), E" < S1v(E, ). Then clearly a(E') E S1(E/) hence a(E') < a(E") = E" by the definition of the partial order in S(E, ). *

Qn-1 _ 1 ) + q 2 . (q-1) (q -1) ... (q 1 -1) We have then n-1 X = ql+2+···+(n-1) + 2 (-l)n-iql+2+···+(i-l) i=l (qn-i+l ... ,::_ _ _-'---,_____:::__ _ ____:. 2 . (q-l)(q -1) ... (q n-1 -1) + (-l)n= 0 (q-l)(q 2 -1) ... (q 1-l) i=l and the formula for {3(n) is proved. ) In order to compute y(n), we add together the recurrence formula for y(n) with the corresponding formula in which n is replaced by n+l. Note that: (-l)nn t· (-l)n 1-1 (nt-l) = (-l)n+l and (-l)n-i (qn-i+l _l)(qn-i; 2-l) ·:· (qn-l) (q-1) (q 2 -1) ...

We shall prove that 40 aA az THE DISCRETE SERIES OF GLn OVER A FINITE FIELD is an isomorphism. Since aA = az ® lA it is sufficient to prove that is an isomorphism, and this would follow if we can prove that aA is an isomorphism whenever A is a field. Assume now that A is a field. We know that dimA He_ 1 (SI(E); A)= (q-1) (q 2 -1) · ·· (qe-1) (cf. 14) and - - - dimA He-1 (S; A)= dimA He-r--1 (SI(E/*); A). dimA Hm-1 (Siv(E, ); A)= (q-1) (q2-l) ... (qe-m_l). (q -m+1 _l) (qe-m+2 _l) ... (qe-1) (cf. *

### Discrete Series of GLn Over a Finite Field. (AM-81) by George Lusztig

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