A compactification of the Bruhat-Tits building.

*(English)*Zbl 0935.20034
Lecture Notes in Mathematics. 1619. Berlin: Springer-Verlag. vii, 152 p. (1995).

The main object of the book under review is the Bruhat-Tits building \(X(G)\) of a connected reductive group \(G\) defined over a local field \(K\). The author regards \(X(G)\) as a \(p\)-adic analogue of a symmetric space, with an eye towards compactifying it. He constructs such a compactification \(\overline X(G)\) being inspired by a polyhedral construction due to Borel-Serre. In this sense, the present construction generalizes that of P. Gérardin [in: Operator algebras and group representations, Vol. I, Monogr. Stud. Math. 17, 208-221 (1984; Zbl 0536.22018)]. The main idea is as follows: first, to compactify an apartment \(A\) of \(X(G)\) (this is done in §2 by accomodating the classical approach of A. Ash, D. Mumford, M. Rapoport, and Y. S. Tai [Smooth compactification of locally symmetric varieties (Math. Sci. Press, Brookline, MA) (1975; Zbl 0334.14007)] to the \(p\)-adic case), and then to use the obtained compactification \(\overline A\) as a “local model” for compactifying \(X(G)\). Namely, viewing \(X(G)\) as the set of equivalence classes of \(G(K)\times A\) modulo a suitable equivalence relation, one can define \(\overline X(G)\) as the set of equivalence classes of \(G(K)\times\overline A\) modulo a natural extension of this relation. This construction is described in detail in §14 (assuming that the residue field of \(K\) is finite) and is illustrated in §15 by the example \(G=\text{SL}_n\).

One should note that the exposition is almost self-contained. In particular, the author reproduces an essential part of the Bruhat-Tits theory. Among his technical innovations it is worth pointing out systematic use of the Néron-Raynaud integral models of algebraic tori (see Chapter 10 of S. Bosch, W. Lütkebohmert, and M. Raynaud [Néron models (Springer, Berlin) (1990; Zbl 0705.14001)]). Another interesting approach is used in the construction of an integral model of \(G\): instead of the representation-theoretic method of Bruhat-Tits, the author employs birational group laws like in Chapter 5 of the above cited book on Néron models.

One should note that the exposition is almost self-contained. In particular, the author reproduces an essential part of the Bruhat-Tits theory. Among his technical innovations it is worth pointing out systematic use of the Néron-Raynaud integral models of algebraic tori (see Chapter 10 of S. Bosch, W. Lütkebohmert, and M. Raynaud [Néron models (Springer, Berlin) (1990; Zbl 0705.14001)]). Another interesting approach is used in the construction of an integral model of \(G\): instead of the representation-theoretic method of Bruhat-Tits, the author employs birational group laws like in Chapter 5 of the above cited book on Néron models.

Reviewer: B.Kunyavskii (Ramat Gan)