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By the _ b b dom. //i. previous case, we have a nonzero map u W r ! r dom. 2 ; That map must be surjective, as can be seen by considering cosocles. Next, the exact b hı i. The bdom. / ! r b hni. 2 ; dom. //i ı u is the desired nonzero map r t u 6 Explicit computations for SL2 For the remainder of the paper, we focus on G D SL2 . 1, we assume that the characteristic of k is not 2. We identify X D Z and XC D Z 0 . 1 agrees with the usual order on Z. In this section, Ä will mean the usual order on Z.

5 below. 3]. In the extreme cases of dominant or antidominant weights, equation (2) actually implies that On exotic and perverse-coherent sheaves b ŠO . w0 /hıw0 i for 2 XC . (3) b The proposition also implies that the composition b ! ON e . /hı i ! r is b appearing in the following nonzero for all 2 X. This is the morphism b ! r statement. 3. N D b CohG b for all is stable under h1i and contains b and r object has finite length. N 2 X. b hni ! r are simple and pairwise nonisomorphic, and every simple object is isomorphic to one of these.

A. edu © Springer International Publishing Switzerland 2015 M. E. N. Achar In this expository article, we will introduce these categories via a relatively elementary approach. But we will also see that they each admit numerous alternative descriptions, some of which play crucial roles in various applications. The paper also touches on selected aspects of their structure theory, and it concludes with detailed calculations for G D SL2 . Some proofs are sketched, but many are omitted. There are no new results in this paper.

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