Kostas Sgouros is sitting on a window ledge in his family's suite at the Watergate Hotel, gazing out the window for one instant of quiet. Then he is off and running, bouncing off the well-upholstered furniture, rolling on the rug, hopping into the lap of a visitor and gazing soulfully into her eyes. At 6 1/2, he is cute as a puppy, but there is no sign at all that he may become a prodigy. That assignment belongs to his brother Dimitris, who is sitting in a corner, wrapped in the dignity of his 12 (almost 13) years and braced for another plunge into the celebrity whirlpool.

"Don't laugh at him," says Dimitris, smiling at Kostas. "I used to be like that. Sometimes, I still am."

Kostas is at the Watergate because the Sgouros family stays together as much as it can. At the moment, the three other members of the family are following Dimitris, who is playing two concerts at Wolf Trap with the National Symphony Orchestra this weekend. In the fall, with the boys' mother, Marianthe, Kostas will go to live in London where Dimitris will study piano at the Royal Academy of Music and begin private lessons with Vladimir Ashkenazy. The boys' father, Soteris Sgouros, will stay with his medical practice in Piraeus, Greece, and visit London whenever he can.

Dimitris Sgouros did not begin playing the piano until he was 7--a late start in the prodigy business--but since then he has compensated spectacularly. In the last two years, since he began winning competitions and giving occasional concerts, he has made headlines in at least six languages. "Une e'toile est ne'e" ("A star is born"), proclaimed the French press after his performance at the Menton Festival. He was pronounced a "nuovo Mozart" in Italy, where he won a competition in 1980, a "Wunderkind" in Germany and a "Wonderkind" in Holland. Marianthe Sgouros keeps copies of the clippings and reads through them, showing fluency in English, French and Italian and an ability to get along in German. "It's not too hard," she says, picking up a clipping in Dutch, "very much like German." Wunderkind--Wonderkind; it's all the same. In any language, we are talking about a startling talent.

Sgouros is so startling, in fact, that he has already become the victim of a poison-pen campaign: letters to the National Symphony Orchestra alleging that he is not 12 years old as has been said, but 17. His mother has the documents to support his claim: a birth certificate signed by the mayor of Piraeus, school records that show him to be an "A" student in the seventh grade, a certificate from the Athens Conservatory that gives his age as 12 and pronounces him qualified to give concerts in public and teach piano at the most advanced levels.

But even without such documents, the suggestion is absurd to anyone who has seen Dimitris Sgouros face-to-face. His cheeks have never been near a razor and will not need one for years. There is nothing extraordinary about him except the reach and strength of his arms, the size and sure movements of his hands, the direct, no-nonsense intelligence of the mind that reveals itself when he puts down a Greek "Popeye" comic book and begins to talk about music:

"From the time I was 3 until I was 7, I loved electricity--electric things, electric noises, the blender and the vacuum cleaner--and we thought I would be an electrician. Then I began the piano: scales--do-re-mi-fa-so-la--and it was beautiful, and I knew I was a musician."

Marianthe Sgouros recalls how it began--by accident. She was out walking with Dimitris one day when they ran into her old piano teacher and, on the spur of the moment, she asked Dimitris if he would like to take lessons. A month later, she says, "he was eating up the piano books one after another and beginning to compose his own music, and the teacher told me, 'I think we should send him to the Conservatory.' After less than a year of lessons, he took the conservatory examination and entered at the sixth-year level."

"I was only half a year older than him," says Dimitris, looking over at Kostas who is playing on the floor, as though the younger brother may become a prodigy before our eyes at any moment. "I was like him."

At 12, testing the outer fringes of a virtuoso career, it is harder for Dimitris to be like Kostas--although it still happens. When Mstislav Rostropovich brought his pet dog, Pooks, to a rehearsal, Dimitris ran off with the dog whenever he had a spare minute. Like Kostas, he is cute, though in a more adult way, and he uses his cuteness for more adult purposes. "Will you teach me Russian?" he asked Nadia Efremov, Rostropovich's assistant and translator, after his debut with the NSO in Carnegie Hall, and she was totally charmed. "Honey," she said, "I will teach you all the Russian in the world."

Sgouros seems to have the necessary equipment for an international superstar--charm and an abundance of sheer, raw talent--but one other basic ingredient is needed: time to grow. And that requires his family's help, without which he might be picked to the bone in a single season by hungry entrepreneurs. A few minutes' conversation with Marianthe Sgouros makes it seem unlikely that this will happen.

"He is not a professional," she says. "Not yet. It is important that he does not perform too much. We try to confine his concerts to school vacation times. This summer he will give only six performances, half of them in Washington." Besides his performances at Wolf Trap with the National Symphony, Sgouros is giving a solo recital to open the University of Maryland Piano Festival next month. He will also give two performances in July at the Newport Music Festival and a performance on Sept. 12 in Karlsruhe, Germany--and that will be the limit of his public performances before he goes back to school. Hundreds of other offers have been turned down. "We try to accept only the most important concerts," says Marianthe Sgouros, "the ones where he will meet the best musicians."

Sgouros "has been a difficult child from his birth," acording to his mother. "Now that the publicity has made him famous, it is not really different; he is still a difficult boy." The family is giving him as much autonomy as it thinks a 12-year-old can handle. "We only follow him," she says, "and we try to protect him--not push him, not hold him back. We leave him free to discuss and to decide what he wants to do."

One decision they are allowing their son to make is his choice of teachers. He has decided to study with Maurizio Pollini after Ashkenazy. The breadth that Pollini can add to his repertoire and the polish he can bring to his technique are probably exactly what Sgouros needs for optimum growth. The fact that Sgouros has chosen such a teacher, rather than a steel-fingered virtuoso, shows a very adult awareness of his present limitations and areas where he needs to grow.

Meanwhile, he is keeping his public appearances to a reasonable number and enjoying himself. Once, he substituted for Lazar Berman, whose Western tour was canceled this spring because he was not allowed to leave the Soviet Union. And at the end of February, in Munich, he substituted for Mstislav Rostropovich, who became ill before a scheduled cello recital. "In Munich," he recalls proudly, "they asked for five encores."