Computing examples

Some of the problems related to affine Weyl groups and Iwahori-Hecke algebras coming up in my work are combinatorially very complex, hence in order to compute non-trivial examples it is often helpful to use a computer. Let me describe some of the problems for which I have written programs to compute examples.

Are local models Cohen-Macaulay?

Local models are schemes over Z_p defined in terms of linear algebra, which model the singularities occuring in the reduction of certain Shimura varieties. It is hence of arithmetic interest to understand their singularities. In some cases, the question whether the local model is Cohen-Macaulay reduces to a purely combinatorial question in terms of the orbit closure relations of a certain stratification of the special fibre.

Since the criterion for being Cohen-Macaulay involves a complicated recursion, it seems very difficult to prove that the local models are Cohen-Macaulay along these lines.

Nevertheless, it is at least possible to check examples: The set of strata can be explicitly described as a certain subset of an affine Weyl group (the so-called μ-admissible set), and the closure relations are given by the Bruhat order. I wrote a computer program which computes all elements up to some length together with their reduced expressions (and hence the Bruhat order) and was able to check that certain low-dimensional local models are indeed Cohen-Macaulay. See U. Görtz, On the flatness of models of certain Shimura varieties of PEL type. Math. Ann. 312 (2001), 689-727.

Another approach to parametrizing the strata is to use a connection to representations of the cyclic quiver. Jan Schröer and I have worked out a combinatorial description in this context which seems indeed better suited than the one coming from the Bruhat order. This enables us to significantly speed up the computations, and hence to compute some more examples. We are still hopeful that we can actually find a proof for a general statement.

The Jordan-Hölder series of certain nearby cycles sheaves

Together with Thomas Haines I have investigated the Jordan-Hoelder series of certain sheaves of nearby cycles on an affine flag manifold. In some cases, these sheaves are related to Shimura varieties and to their Hasse-Weil zeta function. Examples can in principle be computed by considering the function given by taking the trace of Frobenius on the stalks of these sheaves as an element in the Iwahori-Hecke algebra. However, the combinatorics involved is quite complicated (for instance, one has to compute a lot of Kazhdan-Lusztig polynomials), so in any non-trivial case the use of a computer is necessary.

Looking at the results obtained in this way, we found striking combinatorial patterns. Eventually we were able to prove a cohomological explanation for them (which generalizes a theorem of Kazhdan and Lusztig). See U. Görtz, T. Haines, The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties.

Affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties are varieties (at least in the "function field case"; they are locally of finite type, contained in a (partial) affine flag manifold) which are variants of usual Deligne-Lusztig varieties for an affine root system. To an element w of the extended affine Weyl group and a σ-conjugacy class b one can associate an affine Deligne-Lusztig variety X_w(b). But only little is known about their structure, for example there is not even a simple rule saying which of these X_w(b)'s are non-empty.

With the help of a computer program, I was able to check a conjecture by Daniel Reuman for a significant number of cases, and to find a refinement of this conjecture. For more information on this, see U. Görtz, T. Haines, R. Kottwitz, D. Reuman: Dimensions of some affine Deligne-Lusztig varieties.

Visualizing examples

Small-dimensional examples can often be visualized and hence be better understood. I found that often the most appropriate tool to produce pictures of this type is PostScript. See Bill Casselman's book for an introduction to "writing PostScript files by hand". Here are some examples (click on the small pictures to get more typical PostScript files).

Affine Deligne-Lusztig varieties. In the case of rank 2, the results on non-emptiness and the dimension of affine Deligne-Lusztig varieties can be visualized in a picture of the standard apartment of the corresponding Bruhat-Tits building. More explanation and more pictures can be found in the arXiv version of U. Görtz, T. Haines, R. Kottwitz, D. Reuman: Dimensions of some affine Deligne-Lusztig varieties. To appear in Ann. sci. de l'ENS. math.AG/0504443.
This picture shows emptiness/non-emptiness of affine Deligne-Lusztig varieties, and also shows the double Kazhdan-Lusztig cells in this case. In particular, the shrunken Weyl chambers (where the pattern can be described relatively easily ("Reuman's conjecture")) are a cell, and one could hope to find a "closed formula" describing the pattern inside a given cell.
Strong Bruhat order I have written a small script in order to draw the Hasse diagram for the strong Bruhat order, a finer order than the Bruhat order on quotients of finite Weyl groups. This partial order comes up in Torsten Wedhorn's study of F-zips. Torsten has since further expanded this script.
Part of the Bruhat-Tits building for SL_2(Q_2) (a variant of this picture appears in the book Modular forms and special cycles on Shimura curves by S. Kudla, M. Rapoport and T. Yang, see the appendix to ch. VI, p. 19)

Root system paper

A2 paper B2 paper G2 paper

Since PostScript files are just text files, you can easily adapt the root system paper to your purposes: To scale the pictures, replace the 30 in line 3 (/unit {30 mul} def) by a smaller or larger number. To change the shade of grey, replace the .5 in line 4 (.5 setgray) by a smaller (=darker) or greater (=lighter) number (between 0 = black and 1 = white).

Bill Casselman has similar root system paper.