He's the substitute Russian lit teacher at Sasha's school. He has a beard, and the little we know about him makes us think he would probably fit right in with a group of disaffected hipsters with a penchant for quoting philosophy at the local coffee shop. Sasha spies on his classroom while Luzhko is giving a lecture on a Russian short story called "The Nose." Here's how Luzhko breaks it on down for his students:
"What 'The Nose' so vividly demonstrates to us today [...] is that when we blindly believe in someone else's idea of what is right or wrong for us as individuals, sooner or later our refusal to make our own choices could lead to the collapse of the entire political system. An entire country. The world, even." (24.6)
If you're thinking, "Wow... this is some pretty heavy stuff," you're absolutely correct. Luzhko's peddling quite a subversive message here to the good communist kiddos, and catches Sasha at a time when he's pretty vulnerable (he's already pretty much accepted his dad's not coming back, and he's starting to question what really constitutes a "criminal"). Because of this, Sasha regards Luzhko as "suspicious," since he's not using "words you hear on the radio" (24.8).